Individual Dynamical Masses of Ultracool Dwarfs^{* † ‡}

3.1.1. Keck/NIRC2 AO Imaging and Masking

We present here new Keck/NIRC2 AO imaging and non-redundant aperture-masking observations, both in natural guide star and laser guide star modes, in addition to a re-analysis of our own previously published data and publicly available archival data for our sample binaries. Our approach for reducing NIRC2 imaging and obtaining the binary parameters separation, position angle (PA), and flux ratio is well-established from our previous works (Liu et al. 2006, 2008; Dupuy et al. 2009a, 2009b, 2009c, 2010, 2015a). Briefly, we apply standard calibrations (dark subtraction, flat fielding) and then fit an analytic, three-component Gaussian model to each point source in the images. In cases where the binary components are spatially well-separated, we perform PSF-fitting using StarFinder (Diolaiti et al. 2000). In other cases where a third, unsaturated star is in the field (e.g., Gl 569A and Gl 569Bab), we use the third single star as an empirical PSF to fit the binary. Analysis of our NIRC2 masking data was performed using a pipeline similar to previous papers (e.g., Ireland & Kraus 2008; Ireland et al. 2008) and is described in detail in Section 2.2 of Dupuy et al. (2009c). For NIRC2 narrow camera images obtained before 2015 April 13 UT, we use the astrometric solution of Yelda et al. (2010) to correct our measured (x, y) image coordinates for nonlinear optical distortion, using their derived pixel scale of 9.952 ± 0.002 mas pixel⁻¹ and +0252 ± 00.009 correction for the orientation given in the NIRC2 image headers. For narrow camera imaging obtained after 2015 April 13 UT, we use an updated distortion solution from Service et al. (2016) that accounts for the first-ever realignment of the AO system. The post-realignment pixel scale and orientation are 9.971 ± 0.004 mas pixel⁻¹ and +0262 ± 00.020, respectively. For NIRC2 wide camera images, which represent a very small subset of our observations, we used the distortion solution of H. Fu et al. (2012, private communication),⁶ which assumes a pixel scale of 39.686 mas pixel⁻¹ and the Yelda et al. P.A. offset of +0252.

We have made some small improvements to our astrometric analysis compared to our previous work, as described in Dupuy et al. (2016). For data obtained in the vertical angle mode, where the sky rotation of the images relative to the detector axes is constantly changing, we corrected the rotator angles reported in the header to correspond to the midpoint instead of the start of the exposure. We also apply corrections for differential aberration and atmospheric refraction. The refraction correction requires knowledge of the air temperature, pressure, and humidity on Maunakea during our observations, for which we used the weather data archived by CFHT.⁷ We generally adopt errors that are the rms of the measurements from individual dithers at a given epoch. In some cases where we have previously published astrometry, we instead use estimates of the errors from Monte Carlo simulations of fitting artificial binaries.

Table 2 gives our measured astrometry and flux ratios for all Keck AO data used in our orbital analysis. In total there are 339 distinct measurements (unique bandpass and epoch for a given target), where 302 of these are direct imaging and 37 are non-redundant aperture masking. The median Keck separation error is 0.6 mas with 90% of measurements having errors between 0.06–1.9 mas. The median P.A. error is 03, with 90% of measurement errors between 002 and 13. (We caution that there are possible systematic effects, e.g., uncertainty in the distortion solution and tip-tilt jitter, that are difficult to quantify and may impact the few measurements with astrometric errors well below 0.1 mas. The fact that none of these data points show up as outliers in our orbit fitting analysis described later could be due to the fact that not all phases of all orbits are overconstrained by numerous degrees of freedom.) Eight of the imaging measurements are from six unpublished archival data sets. For HD 130948BC we used data sets from 2005 February 25 UT (PI Prato) and 2011 March 25 UT (PI Bowler). For Gl 569Bab we used data sets from 2003 April 15 UT (PI Simon), 2004 August 10 UT (PI Kulkarni), and 2011 March 25 UT (PI Bowler). For LSPM J1735+2634AB we used a JHK data set from 2007 August 1 UT (PI N. Law). Fourteen other measurements are from our re-analysis of 10 previously published data sets in the archive: LP 349-25, 2MASS J0920+3517AB, 2MASS J0746+2000AB, HD 130948BC, 2MASS J1847+5522AB, and 2MASS J2206−2047AB (PI Ghez; Konopacky et al. 2010) and SDSS J2052−1609AB (PI C. Gelino; Bardalez Gagliuffi et al. 2015).

Table 2.  Relative Astrometry from Keck/NIRC2 Adaptive Optics Imaging and Masking

Observation Date		Separation	PA	Δm	Bandpass	Nframes	Notes
(UT)	(MJD)	(mas)	(°)	(mag)
LP 349-25AB (Nep = 8, Δt = 3.99 years)
2008 Jan 16	54481.25	137.25 ± 0.24	208.07 ± 0.06	0.335 ± 0.016	${K}_{S}$	10	I
2008 Jun 30	54647.55	114.88 ± 0.25	194.94 ± 0.12	0.285 ± 0.011	${K}_{S}$	9	I
2008 Jun 30	54647.56	114.67 ± 0.23	195.13 ± 0.08	0.303 ± 0.004	H	7	I
2008 Jun 30	54647.56	114.5 ± 0.6	195.20 ± 0.17	0.345 ± 0.011	J	11	I
2008 Aug 20	54698.61	105.71 ± 0.10	189.90 ± 0.05	0.353 ± 0.010	${K}_{S}$	12	I
2008 Sep 9	54718.58	102.17 ± 0.14	187.62 ± 0.09	0.243 ± 0.005	L'	12	I
2009 Sep 28	55102.56	71.20 ± 0.30	98.4 ± 0.6	0.230 ± 0.030	${K}_{S}$	14	I
2009 Dec 15	55180.37	83.6 ± 0.4	81.40 ± 0.30	0.37 ± 0.06	K	14	I
2010 May 22	55338.62	112.76 ± 0.18	59.93 ± 0.09	0.397 ± 0.019	K	13	Ia
2012 Jan 14	55940.25	131.4 ± 0.7	19.52 ± 0.12	0.258 ± 0.011	K	15	I

Notes. In the notes column, "I" indicates an observation done with direct imaging and "M" indicates non-redundant aperture masking.

^aThis denotes a previously published data set that we obtained from the public NIRC2 archive. ^bThis denotes a previously unpublished data set that we obtained from the public NIRC2 archive.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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3.1.2. HST/ACS-WFC Monitoring

3.1.3. Other Published and Archival HST Imaging

We obtained high-precision unresolved astrometry of our sample to determine parallaxes and proper motions, and to constrain the photocenter motion due to the binary orbits. We present here an updated analysis of our data from the Hawaii Infrared Parallax Program that uses the CFHT facility infrared camera WIRCam (Puget et al. 2004). Our observing strategy and custom astrometry pipeline are described in detail in Dupuy & Liu (2012). Briefly, for a given target we obtain data sets on multiple nights each season, with each data set typically comprising 20–30 dithered frames in the J band, or in a narrow K-band filter (${K}_{{\rm{H}}2}$) if the target would saturate in J. Queue observing constraints at CFHT are used to require good seeing and that all J-band observations occur near transit, i.e., at similar airmass, to guard against systematic astrometric errors due to differential chromatic refraction (DCR). We measure Gaussian-windowed centroids by running Sextractor v2.19.5 (Bertin & Arnouts 1996) on the detrended images provided by CFHT. We cross-match detections at a given epoch, using our custom distortion solution and simple linear transformations, and adopt the standard error on the mean as the astrometric measurement error for any given star at each epoch. We then solve for the linear transformations between epochs, iteratively solving for and masking high proper motion and/or parallax sources. We finally cross-match to an astrometric reference catalog (2MASS or SDSS-DR9) to provide absolute calibration of the linear terms and then solve for the proper motion and parallax of every star in the field.

Unlike our previous work, we do not use a stand-alone analysis of our CFHT data to compute the parallax and proper motion of our target binaries. This approach was used by Dupuy & Liu (2012), where we then applied evolutionary model-based corrections to account for the expected orbital motion in the CFHT photocenter. As we have continued monitoring our sample binaries from 2007 or 2008 up to the present, we now have data spanning a much longer fraction of many of our targets' orbital periods. It therefore becomes even more important to freely fit for photocenter motion (as described in Section 4), without imposing assumptions about mass–magnitude relations from evolutionary models or potentially unresolved multiplicity. Indeed, the fact that we are now in a position to detect significant photocenter motion allows us to place strong empirical constraints on the relationship between mass and magnitude, as we have recently demonstrated with SDSS J1052+4422AB (Dupuy et al. 2015b), and possibly detect higher-order multiplicity. We have also made some small changes to improve the performance of our pipeline compared to Dupuy & Liu (2012), such as using the mean instead of the median measured position of a given source at each epoch. In addition, when computing the correction from relative to absolute parallax and proper motion using the Besançon model of the Galaxy (Robin et al. 2003), we now exclude model stars with proper motions larger than the cutoff value we use in our iterative solution of the epoch-to-epoch linear terms.

Table 4 gives all of the absolute astrometry from CFHT for the sample of binaries for which we have orbit-monitoring data from Keck and HST. Many of these CFHT observations were originally published in Dupuy & Liu (2012), but the data presented here are nonetheless distinct from what was published previously since we no longer attempt to remove any photocenter motion from the reported absolute astrometry.

Table 4.  Integrated-light Astrometry from CFHT/WIRCam

Observation Date		R.A.	Decl.	${\sigma }_{{\rm{R}}.{\rm{A}}.}\cos \delta$	σDecl.	Airmass	Seeing
(UT)	(MJD)	(deg)	(deg)	(mas)	(mas)		(arcsec)
LP 349-25AB (Nep = 20, Δt = 8.26 years)
2008 Aug 9	54687.5662	006.98451510	+22.32547171	2.4	2.6	1.001	0.61
2008 Sep 6	54715.4857	006.98451751	+22.32546921	3.0	8.6	1.001	0.48
2008 Oct 8	54747.4233	006.98451811	+22.32545820	2.8	3.3	1.008	0.53
2008 Oct 9	54748.3920	006.98451801	+22.32545770	2.7	3.5	1.002	0.75
2008 Nov 16	54786.2900	006.98451972	+22.32544801	1.9	3.2	1.002	0.90
2009 Jul 29	55041.6143	006.98463346	+22.32542572	2.6	4.0	1.006	0.84
2009 Aug 4	55047.6397	006.98463611	+22.32542544	3.1	3.6	1.063	0.56
2009 Aug 8	55051.5876	006.98463564	+22.32542301	2.0	2.3	1.006	0.68
2009 Aug 25	55068.5222	006.98463634	+22.32542258	2.8	2.1	1.001	0.68
2009 Oct 21	55125.3717	006.98463934	+22.32540909	2.8	2.5	1.001	0.71
2009 Nov 6	55141.3542	006.98463957	+22.32540230	8.4	8.7	1.016	0.79
2009 Dec 22	55187.2217	006.98464710	+22.32538996	1.6	1.8	1.010	0.58
2010 Aug 15	55423.5592	006.98475726	+22.32537588	1.4	2.4	1.002	0.58
2010 Sep 15	55454.4572	006.98476077	+22.32536973	3.1	2.8	1.003	0.47
2010 Oct 22	55491.3577	006.98475905	+22.32536073	5.2	3.8	1.002	0.63
2011 Jul 19	55761.6019	006.98487661	+22.32533064	3.3	1.5	1.012	0.53
2011 Jul 25	55767.5641	006.98487564	+22.32533050	3.1	4.2	1.039	0.69
2016 Aug 17	57617.5512	006.98549184	+22.32509917	4.6	5.2	1.002	0.46
2016 Sep 10	57641.4343	006.98549407	+22.32509416	3.6	5.6	1.038	0.57
2016 Nov 13	57705.2584	006.98549810	+22.32507409	18.4	12.0	1.040	0.51

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Although we have applied our MCMC analysis to all of our astrometric data, not all binaries will have orbit determinations sufficiently reliable for astrophysical use. The primary quality metric for assessing our orbit fits should be the precision with which they constrain the system mass, as very large mass uncertainties will be of little use for constraining models in the following analysis. In addition, if orbital parameters are poorly determined, then the resulting posteriors will be strongly influenced by our adopted priors. Although we have attempted to choose priors that are as uninformed as possible, we should not rely on them to constrain physical parameters (P, a, e) that are not sufficiently constrained by our data. Finally, some of our orbit determinations indicate that our observations span only a small fraction of the full orbital period. Therefore, we also consider this as a possible indicator of whether our data has provided reliable constraints on orbital parameters.

Table 7 provides a summary of the orbit quality metrics that we use to assess the reliability of our orbit fits. The metrics $\delta \mathrm{log}{M}_{\mathrm{tot}}$ and δe are computed as the difference between the maximum and minimum credible interval (1σ) of the total mass at fixed distance and eccentricity, respectively, and the metric ${\rm{\Delta }}{t}_{\mathrm{obs}}/P$ is the fraction of the orbital period (median of the posterior) covered by our resolved astrometry. We now consider our orbit fits, beginning with the apparently least reliable, and moving toward binaries with better reliability metrics until we finally determine which orbits to carry forward in our analysis.

Table 7.  Summary of Orbit Quality Metrics

		Orbit Quality Metrics
Name	SpT	${\rm{\Delta }}\mathrm{log}{M}_{\mathrm{tot}}$	Δe	Δtobs/P	P	${M}_{\mathrm{tot}}$
	(A+B)	(dex)			(years)	(${M}_{\mathrm{Jup}}$)
Orbit Determinations Used in Our Analysis
LP 349-25AB	M7+M8	0.0035	0.0037	0.978	7.698 ± 0.014	${166}_{-7}^{+6}$
LP 415-20AB	M6+M8	0.0586	0.0233	0.673	14.82 ± 0.24	${248}_{-29}^{+26}$
SDSS J0423−0414AB	L6.5+T2	0.0093	0.0147	0.659	12.30 ± 0.06	83 ± 3
2MASS J0700+3157AB	L3+L6.5	0.0035	0.0128	0.431	23.9 ± 0.5	${141}_{-5}^{+4}$
LHS 1901AB	M7+M7	0.0029	0.0017	0.497	16.21 ± 0.08	${213}_{-10}^{+9}$
2MASS J0746+2000AB	L0+L1.5	0.0050	0.0007	0.760	${12.736}_{-0.029}^{+0.031}$	163 ± 5
2MASS J0920+3517AB	L5.5+L9	0.0098	0.0130	1.822	7.258 ± 0.009	187 ± 11
2MASS J1017+1308AB	L1.5+L3	0.0119	0.0196	0.647	${18.60}_{-0.23}^{+0.22}$	${156}_{-18}^{+14}$
2MASS J1047+4026AB	M8+L0	0.0048	0.0027	2.136	${6.562}_{-0.026}^{+0.029}$	${178}_{-12}^{+11}$
SDSS J1052+4422AB	L6.5+T1.5	0.0096	0.0046	1.048	${8.608}_{-0.024}^{+0.025}$	${90}_{-5}^{+4}$
Gl 417BC	L4.5+L6	0.0114	0.0064	0.846	15.65 ± 0.08	${99.2}_{-3.3}^{+3.0}$
LHS 2397aAB	M8+...	0.0072	0.0063	1.178	14.37 ± 0.05	${159}_{-8}^{+7}$
Kelu-1AB	L2+L4	0.0053	0.0098	0.667	24.98 ± 0.19	${180}_{-26}^{+22}$
2MASS J1404−3159AB	L9+T5	0.0445	0.0106	0.536	${16.52}_{-0.22}^{+0.21}$	${120}_{-13}^{+11}$
HD 130948BC	L4+L4	0.0023	0.0034	1.063	10.009 ± 0.010	${115.4}_{-2.1}^{+2.2}$
Gl 569Bab	M8.5+M9	0.0040	0.0020	5.006	2.3707 ± 0.0005	138 ± 7
2MASS J1534−2952AB	T4.5+T5	0.0060	0.0055	0.719	${20.35}_{-0.06}^{+0.05}$	99 ± 5
2MASS J1728+3948AB	L5+L7	0.0089	0.0277	0.337	${40.8}_{-1.2}^{+1.6}$	${140}_{-8}^{+7}$
LSPM J1735+2634AB	M7.5+L0	0.0017	0.0075	0.286	${21.65}_{-0.23}^{+0.24}$	187 ± 7
2MASS J2132+1341AB	L4.5+L8.5	0.0192	0.0094	0.576	${10.74}_{-0.17}^{+0.16}$	${128}_{-8}^{+7}$
2MASS J2140+1625AB	M8+L0.5	0.0101	0.0146	0.612	24.45 ± 0.25	${183}_{-17}^{+14}$
2MASS J2206−2047AB	M8+M8	0.0140	0.0160	0.578	${23.96}_{-0.21}^{+0.23}$	${188}_{-17}^{+16}$
DENIS J2252−1730AB	L4+T3.5	0.0176	0.0182	0.945	${8.822}_{-0.027}^{+0.026}$	101 ± 7
Marginal Orbit Determinations
DENIS J1228−1557AB	L5.5+L5.5	0.1201	0.0623	0.292	${50}_{-7}^{+5}$	${106}_{-19}^{+16}$
2MASS J1847+5522AB	M6+M7	0.0346	0.1363	0.216	${45}_{-3}^{+5}$	${270}_{-31}^{+26}$
Poorly Constrained Orbits
2MASS J0850+1057AB	L6.5+L8.5	0.1167	0.1118	0.310	${48}_{-6}^{+7}$	54 ± 8
SDSS J0926+5847AB	T3.5+T5	0.3815	0.3761	0.682	${12.9}_{-1.9}^{+1.3}$	${38}_{-18}^{+10}$
SDSS J1021−0304AB	T0+T5	0.0600	0.1435	0.101	${86}_{-17}^{+13}$	${52}_{-7}^{+6}$
SDSS J1534+1615AB	T0+T5.5	0.0881	0.4296	0.175	${58}_{-24}^{+39}$	${46}_{-7}^{+6}$
2MASS J1750+4424AB	M6.5+M8.5	0.1963	0.1523	0.038	${210}_{-60}^{+40}$	${190}_{-50}^{+40}$
SDSS J2052−1609AB	L8.5+T1.5	0.2291	0.2069	0.195	${44}_{-14}^{+10}$	${69}_{-20}^{+14}$

Note. ${\rm{\Delta }}\mathrm{log}{M}_{\mathrm{tot}}$ is defined as the difference between the maximum and minimum values of the 68.3% (1σ) credible interval of the total mass posterior distribution without including the uncertainty in the distance. Likewise, Δe is the difference between the maximum and minimum values of the 1σ credible interval of the eccentricity posterior distribution. Δt_obs/P is the time baseline of the resolved astrometry used in the orbit fit divided by the median of the orbital period posterior distribution. For convenience, we also list here the median and 1σ credible intervals of the orbital period and the total mass including the uncertainty on the distance.

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SDSS J0926+5847AB has the worst mass precision ($\delta \mathrm{log}{M}_{\mathrm{tot}}=0.38$ dex) and eccentricity constraint (δe =0.38). Even though it has a good time baseline (${\rm{\Delta }}{t}_{\mathrm{obs}}/P=0.68$), the actual coverage of the orbit on the sky is poor, partly due to the fact that it is seen nearly edge-on ($i=91\buildrel{\circ}\over{.} {7}_{-0.8}^{+0.6}).$ The next worse mass precisions are ${\rm{\Delta }}\mathrm{log}{M}_{\mathrm{tot}}=0.23$ dex for SDSS J2052−1609AB and 0.20 dex for 2MASS J1750+4424AB. The latter has the smallest time baseline of any binary here (${\rm{\Delta }}{t}_{\mathrm{obs}}/P=0.04$), owing to its very long, quite uncertain orbital period (${210}_{-60}^{+40}$ years), although it only has a modestly large δe = 0.15. SDSS J2052−1609AB has a better time baseline (${\rm{\Delta }}{t}_{\mathrm{obs}}/P=0.19$) but a poorly constrained eccentricity (δe = 0.21). 2MASS J0850+1057AB has a relatively poor mass precision (${\rm{\Delta }}\mathrm{log}{M}_{\mathrm{tot}}=0.12$ dex) as well as a poorly constrained eccentricity (δe = 0.11) despite a marginally acceptable time baseline (${\rm{\Delta }}{t}_{\mathrm{obs}}/P=0.31$). We do not consider any of the above orbits sufficiently reliable to be used in the following analysis. SDSS J1534+1615AB and SDSS J1021−0304AB nominally have much better mass precision than any of the above ($\delta \mathrm{log}{M}_{\mathrm{tot}}=0.09$ dex and 0.06 dex, respectively). However, both have poor time baselines with ${\rm{\Delta }}{t}_{\mathrm{obs}}/P=0.17$ and 0.10, respectively, and poorly constrained eccentricities with δe = 0.43 and 0.14, respectively. (For reference, our input prior on eccentricity alone would correspond to δe = 0.68.) We therefore also chose not to use these orbits.

DENIS J1228−1557AB and 2MASS J1847+5522AB are both marginal cases. DENIS J1228−1557AB has a modestly large mass error ($\delta \mathrm{log}{M}_{\mathrm{tot}}=0.12$ dex) but a better constrained eccentricity (δe = 0.06) and longer observational baseline (${\rm{\Delta }}{t}_{\mathrm{obs}}/P=0.29$). On the other hand, 2MASS J1847+5522AB has good mass precision ($\delta \mathrm{log}{M}_{\mathrm{tot}}=0.035$ dex) but a poorly constrained eccentricity (δe = 0.14) and an observational baseline of only ${\rm{\Delta }}{t}_{\mathrm{obs}}/P=0.22$. We conservatively choose to place our cutoffs in orbit quality metrics to exclude these two marginal cases. The next worse mass precision in our sample is LP 415-20AB ($\delta \mathrm{log}{M}_{\mathrm{tot}}=0.06$ dex), but it appears to have a reliably determined orbit, with δe = 0.023 and most of the orbital period covered by observations (${\rm{\Delta }}{t}_{\mathrm{obs}}/P=0.67$). The worst eccentricity constraint for any of the remaining binaries is for 2MASS J1728+3948AB (δe = 0.028), but it has a much better mass precision ($\delta \mathrm{log}{M}_{\mathrm{tot}}=0.0089$ dex) and better orbital coverage (${\rm{\Delta }}{t}_{\mathrm{obs}}/P=0.34$) compared to our excluded orbits. LSPM J1735+2634AB has the worst orbital coverage of our remaining sample (${\rm{\Delta }}{t}_{\mathrm{obs}}/P=0.29$), but its orbit is very well determined ($\delta \mathrm{log}{M}_{\mathrm{tot}}=0.0017$ dex, δe = 0.0075), thanks to our observations serendipitously bracketing its periastron passage.

In summary, we do not use the orbit determinations for 2MASS J0850+1057AB, SDSS J0926+5847AB, DENIS J1228−1557AB, 2MASS J1750+4424AB, 2MASS J1847+5522AB, and SDSS J2052−1609AB in our following analysis. The basis for their exclusion is poor mass precision, poor observational coverage of the orbit, and/or poorly constrained eccentricity that could make our results overly dependent on our uniform eccentricity prior. The remaining sample comprises 23 binaries with orbit quality metrics ranging from $\delta \mathrm{log}{M}_{\mathrm{tot}}\,=0.0017\mbox{--}0.06$ dex, δe = 0.0007–0.028, and ${\rm{\Delta }}{t}_{\mathrm{obs}}/P\,=0.29\mbox{--}5.01.$

Some of the binaries in our sample have published orbital parameter determinations, either from work of our own or others. By far, the largest overlap of our 31 binary sample is with 14 binaries from Konopacky et al. (2010), followed closely by our own work (13 binaries). In most of these overlapping cases, our new orbit determinations agree with and improve upon previously published work as expected given our longer time baseline and more numerous epochs. For example, compared to Konopacky et al. (2010), our orbital period uncertainties are much smaller, with a median difference in errors of a factor of 7. We now discuss the cases where our new results for astrophysical parameters (i.e., mass, period, and eccentricity, but not viewing angles) differ significantly (≳2σ) from published work.

Gl 569Bab has had a number of published orbit determinations, and despite being a very well-studied system, the reported semimajor axes of the orbit have varied significantly compared to the reported errors. As discussed in Section 3.1 of Dupuy et al. (2010), this is likely mostly due to astrometric calibration issues in data sets that used an early generation of cameras behind the Keck AO system (KCAM and SCAM; Lane et al. 2001), which we did not use here or in our previous work. Our analysis here uses some archival NIRC2 imaging not included in our previous work, which improves the coverage of this short period orbit (P = 2.3707 ± 0.0005 years), and results in a semimajor axis of 93.64 ± 0.14 mas. This sits in between values derived in our prior work (${95.6}_{-1.0}^{+1.1}$ mas Dupuy et al. 2010) and values published by others that largely relied on KCAM and SCAM data: 91.8 ± 1.0 mas (Zapatero Osorio et al. 2004), 90.4 ± 0.7 mas (Simon et al. 2006), and 90.8 ± 0.8 mas (Konopacky et al. 2010). In spite of these different semimajor axes, the dynamical total mass (138 ± 7 ${M}_{\mathrm{Jup}}$ here) varies by <1σ because the error is dominated by the Hipparcos parallax uncertainty (σ_π/π = 0.016), not our semimajor axis measurement (σ_a/a = 0.0016). The improved precision of our new orbit fit is due to more measurements (19 from astrometrically well-calibrated Keck/NIRC2 or HST imaging) that now span 11.87 years (i.e., 5.006 orbital periods).

HD 130948BC has a very well-determined orbit, so small differences in orbital parameters are more statistically significant. In our most recent previous analysis we found $e=0.176\pm 0.006$ (Dupuy & Liu 2011) compared to 0.1627 ± 0.0017 here, a 2.1σ difference. This difference is not astrophysically significant, and perhaps it is simply a statistical fluctuation due to the significant improvement in other orbital parameters (e.g., both a and P are improved by nearly a factor of 20 compared to our original orbit in Dupuy et al. 2009b). Both our current and past values are consistent at <2σ with the eccentricity of 0.16 ± 0.01 from Konopacky et al. (2010).

Our orbit for LP 415-20AB agrees with our previous work but disagrees somewhat with the analysis of Konopacky et al. (2010). This is discussed in detail in Section 2 of Dupuy & Liu (2011) as likely being jointly due to the small number of degrees of freedom in the Konopacky et al. (2010) analysis (1 dof) and one of their measurements being a significant outlier with respect to the rest of the available data. Compared to their orbital parameters, our period of 14.82 ± 0.24 years is longer (compared to 11.5 ± 1.2 years), our semimajor axis of ${96.5}_{-1.4}^{+1.1}$ mas is smaller (compared to 108 ± 24 mas), and our eccentricity of ${0.706}_{-0.012}^{+0.011}$ is smaller (compared to 0.9 ± 0.1). Our inferred total dynamical mass is quite discrepant with that of Konopacky et al. (2010), given that their fitted distance of 21 ± 5 pc (derived by combining astrometry and radial velocities) is 2.7σ different from our parallactic distance of 38.6 ± 1.1 pc. This is likely due both to our different orbit determinations and the fact that their radial velocity measurement (ΔRV = −0.7 ± 1.4 km s⁻¹) disagrees with the prediction from our relative orbit (ΔRV = −2.96 ± 0.16 km s⁻¹). With 12 measurements spanning 9.97 years, our new orbit fit has many more degrees of freedom (17 dof) for the relative orbit fit than previous work, making the resulting orbit parameters more robust and more precise by a factor of 5–17.

Our orbit for 2MASS J1534−2952AB agrees well with Konopacky et al. (2010) but disagrees with Liu et al. (2008). The cause of the discrepancy with Liu et al. (2008) is the choice of eccentricity prior. In that work we adopted a prior of p(e) = e, which deweights smaller eccentricities and rules out circular orbits entirely, whereas we adopt here a more conservative uniform prior p(e) = 1. Our eccentricity posterior here piles up at zero with a 2σ interval of e = 0.000–0.014, and a and P are both correlated with e in the sense that a larger e corresponds to smaller a and P. According to Figure 6 of Liu et al. (2008), that posterior distribution of P as e approaches zero would agree with our orbital period of P = 20.29 ± 0.07 years. According to Figure 8 of Liu et al. (2008), our values of P and a = 213.7 ± 0.5 mas agree well with that P–a posterior distribution in spite of the difference in eccentricity priors. However, both Liu et al. (2008) and Konopacky et al. (2010) used the parallax of 73.6 ± 1.2 mas from Tinney et al. (2003) to convert their angular semimajor axes into physical units and thereby compute dynamical masses. As we discuss in Appendix B.17, our parallax of 63.0 ± 1.1 mas leads to very different dynamical masses than in previous work.

2MASS J2206−2047AB also has an eccentricity posterior that piles up at zero, which differs from the results of our previous work on this system. In Dupuy et al. (2009a) we used a prior of p(e) = e that suppressed low eccentricity orbit solutions and resulted in e = 0.25 ± 0.08. Our new posterior has a 2σ interval of e = 0.000–0.027. As a result of this difference, Dupuy et al. (2009a) found very different values for semimajor axis and period ($a={213}_{-18}^{+24}$ mas, $P={35}_{-5}^{+6}$ years) compared to our new analysis (a = 167.7 ± 0.5 mas, $P={23.96}_{-0.21}^{+0.23}$ years). However, thanks to the strong correlation between a and P, our new values follow the posterior distribution of Dupuy et al. (2009a) as shown in their Figure 6. Therefore, the resulting total dynamical mass was 184 ± 4 ${M}_{\mathrm{Jup}}$ from Dupuy et al. (2009a) but is ${188.3}_{-3.1}^{+2.9}$ ${M}_{\mathrm{Jup}}$ here (assuming a fixed distance of 28.0 pc for the purpose of this comparison). Therefore, the choice of eccentricity prior does not change the best-fit total mass, although it would have affected the uncertainty in the mass, where the uniform e prior used here is more conservative.

SDSS J2052−1609AB has a preliminary orbit determination from Bardalez Gagliuffi et al. (2015) based on combining their data with published astrometry from Stumpf et al. (2011) and three epochs of our data retrieved from the NIRC2 archive. Their posterior eccentricity distribution ($e={0.014}_{-0.010}^{+0.023}$) is significantly different from what we find in our analysis ($e={0.20}_{-0.11}^{+0.09}$, with a 2σ interval of 0.08–0.50). Their quoted uncertainties in other parameters are also generally much smaller than ours, e.g., they find ${M}_{\mathrm{tot}}={86.2}_{-1.8}^{+3.9}$ ${M}_{\mathrm{Jup}}$ while we find ${M}_{\mathrm{tot}}={69}_{-20}^{+14}$ ${M}_{\mathrm{Jup}}$, despite our analysis using more data over a longer time baseline. We also performed a cursory analysis of their astrometry using our MCMC fitter and were unable to reproduce their posterior distributions. Therefore, we speculate that the difference in our results is most likely due to differing MCMC analysis methods. They report that acceptance fractions for their single Metropolis-Hastings sampled chain were typically 0.5%–1%, whereas our analysis using the affine-invariant sampler of emcee had much higher acceptance fractions of 8.7% (Table 6). Their MCMC also included distance as an eighth parameter in addition to the seven orbit parameters that are constrained by the relative astrometry. Since distance is not constrained by the relative astrometry, it appears that its use as a parameter was intended to marginalize over the uncertainty in the Dupuy & Liu (2012) parallax. It is therefore curious that the posterior distribution on distance from Bardalez Gagliuffi et al. (2015) had smaller errors (${30.7}_{-0.4}^{+0.2}$ pc) than the input parallactic distance of 29.5 ± 0.7 pc. Regardless, it is not clear how this would explain the small uncertainties on the orbital parameters or why their MCMC analysis apparently avoided the part of parameter space preferred by our emcee analysis. Our new orbit fit is based on eight epochs, rather than six epochs, spanning 8.58 years instead of 4.55 years, in addition to the fact that our joint Keck+CFHT analysis properly marginalizes over the uncertainty in orbital photocenter motion when determining the parallax. Therefore, we conservatively choose to use our own orbit-fitting results in the following analysis.

Kelu-1AB has an unrefereed orbit determination by Stumpf et al. (2008) based on nine epochs of astrometry spanning 2.84 years. Our orbit based on 13 epochs spanning 16.66 years agrees at ≲1σ with their eccentricity (e = 0.82 ± 0.10) but not their semimajor axis ($a={339}_{-66}^{+129}$ mas) or orbital period ($P={38}_{-6}^{+8}$ years). We find $a={227.9}_{-1.1}^{+0.9}$ mas and P = 24.98 ±0.19 years. Assuming a fixed distance of 20.8 pc for both orbits, we find a total mass of ${M}_{\mathrm{tot}}=180.1\pm 1.1$ ${M}_{\mathrm{Jup}}$ compared to their ${M}_{\mathrm{tot}}={244}_{-76}^{+156}$ ${M}_{\mathrm{Jup}}$, which agrees within 1σ because of their large uncertainties.

In Table 8 we provide a comparison of the parallaxes we measure here with other values in the literature, including our own past works (Dupuy & Liu 2012; Dupuy et al. 2015b). Our previously published parallaxes are not statistically independent of the values given here, but they were derived from somewhat different analysis methods. Namely, in Dupuy & Liu (2012) we did not marginalize over the uncertainty in the orbital parameters or mass ratio in the same way as we have done here, and we used a slightly different method for computing the correction from relative to absolute parallax. It is therefore not surprising that one of the largest discrepancies is for 2MASS J0920+3517AB (2.0σ different from our published parallax), because we previously assumed a model-based mass ratio estimated from the measured flux ratio, but as we discuss in Section 7 and Appendix B.7 this system is likely a hierarchical triple where the fainter component is actually much more massive than the brighter component. The only other ≳2σ difference is for 2MASS J1728+3948AB, where our new data spans 9.1 years (instead of 3.3 years) and with somewhat better parallax phase coverage resulting in a parallax that is 6% (2.5σ) smaller. We suggest this is likely a natural statistical variation given the new better constraints on proper motion in both R.A. and decl. (the parallax amplitude in decl. is almost as large as R.A. for this object).

Among other literature parallaxes, the largest discrepancy is for 2MASS J1534−2952AB (3.9σ). As we have previously discussed in Dupuy & Liu (2012), this is likely due to the uncertainty in the Tinney et al. (2003) value of 73.6 ± 1.2 mas being somewhat underestimated. Our value of 63.0 ± 1.1 mas significantly changes the inferred dynamical mass since ${M}_{\mathrm{tot}}\propto {d}^{3}$, resulting in a factor of 1.6 increase in mass. The next largest discrepancies are 1.3σ−1.9σ for LP 349-25AB, SDSS J0423−0414AB, 2MASS J0700+3157AB, and 2MASS J0746+2000AB and the remaining 19 comparisons agree within ≤0.6σ. Two binaries have parallaxes in the literature more precise than our CFHT values. Kelu-1AB has a parallax of 53.6 ± 2.0 mas from Dahn et al. (2002) and 51.75 ± 1.16 mas from Weinberger et al. (2016) that are both more precise but likely less conservative than our value of 49.8 ± 2.2 mas because we marginalize over photocenter orbital motion. All of these values agree with each other at 0.8σ–1.3σ, but using the different parallaxes results in quite different dynamical masses, and we therefore exclude Kelu-1AB from our further mass analysis (see Appendix B.13 for a discussion). 2MASS J0746+2000AB has a remarkably precise parallax of 81.9 ± 0.3 mas from the USNO optical program (Dahn et al. 2002). An updated USNO analysis that accounts for the photocenter orbital motion of 2MASS J0746+2000AB (Harris et al. 2015) results in a parallax of 81.24 ± 0.25 mas (H. Harris 2015, private communication). Given the longer time baseline and higher precision of the USNO parallax, we use their value in our following analysis.

Most binaries in our sample have spectral types derived through spectral decomposition from Dupuy & Liu (2012), using integrated-light NIR spectra and resolved NIR photometry. For most of these we simply adopt the same spectral types as in that work, but a few binaries were not in that sample or have updated resolved photometry that warrants new analysis. We use the same method described in Section 5.2 of Dupuy & Liu (2012). Briefly, we pair spectra from a template library and find the optimal scaling ratio for each pair that best matches the observed integrated-light spectrum. We compute synthetic photometry for these pairs and then cull all pairings that do not agree with the observed resolved photometry, p(χ²) < 0.05. We then examine the remaining pairs, ranked by how well they match the observed spectrum, and assign types and errors on types that best represent the results. A summary of all available magnitudes for our sample binaries, both integrated-light and resolved, is given in Table 9. We report magnitudes on both the 2MASS and MKO photometric systems if data exist in either system by using the empirical relations between photometric system conversion and absolute magnitude that we derived in Appendix A.1.

Table 9.  Apparent Magnitudes for the Sample

Filter	Integrated	${\rm{\Delta }}m={m}_{{\rm{B}}}-{m}_{{\rm{A}}}$	mA	mB	References
LP 349-25AB
MKO J	10.563 ± 0.022	0.350 ± 0.030	11.155 ± 0.025	11.505 ± 0.028	Cutr03, Dupu10, Dupu12
MKO H	10.005 ± 0.023	0.326 ± 0.011	10.607 ± 0.023	10.933 ± 0.024	Cutr03, Dupu10, Dupu12
MKO K	9.540 ± 0.017	0.307 ± 0.008	10.150 ± 0.017	10.457 ± 0.018	Cutr03, Dupu10, Dupu12
2MASS J	10.615 ± 0.025	0.346 ± 0.037	11.208 ± 0.029	11.555 ± 0.032	Cutr03, Dupu10, Dupu12, Dupu17
2MASS H	9.976 ± 0.023	0.320 ± 0.012	10.580 ± 0.024	10.900 ± 0.024	Cutr03, Dupu10, Dupu12, Dupu17
2MASS KS	9.569 ± 0.017	0.318 ± 0.007	10.174 ± 0.017	10.492 ± 0.017	Cutr03, Dupu10

References. Burg03b—Burgasser et al. (2003b), Burg06b—Burgasser et al. (2006b), Burg11—Burgasser et al. (2011), Chiu06—Chiu et al. (2006), Clos03—Close et al. (2003), Cutr03—Cutri et al. (2003), Dupu09a—Dupuy et al. (2009a), Dupu09b—Dupuy et al. (2009b), Dupu09c—Dupuy et al. (2009c), Dupu10—Dupuy et al. (2010), Dupu12—Dupuy & Liu (2012), Dupu14—Dupuy et al. (2014), Dupu15—Dupuy et al. (2015b), Dupu17—this work, Gizi03—Gizis et al. (2003), Knap04—Knapp et al. (2004), Kono10—Konopacky et al. (2010), Lane01—Lane et al. (2001), Legg00—Leggett et al. (2000), Legg02a—Leggett et al. (2002), Liu08—Liu et al. (2008), Reid06a—Reid et al. (2006a), Stum11—Stumpf et al. (2011).

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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LP 415-20AB and 2MASS J1047+4026AB were not included in the Dupuy & Liu (2012) sample. LP 415-20AB has an integrated-light optical spectral type of M7.5 (Gizis et al. 2000a), but we find that primary templates as early as M5 and as late as M7 provide good matches to the infrared SpeX SXD spectrum and resolved photometry, while the best secondary templates range from M7.5–M8.5. Therefore, we adopt types of M6.0 ± 1.0 for LP 415-20A and M8.0 ± 0.5 for LP 415-20B. 2MASS J1047+4026AB has an integrated-light optical type of M8 (Gizis et al. 2000b), and we find types of M8.0 ± 0.5 and L0.0 ± 1.0 in our spectral decomposition. In this work we have added new flux ratios in the CH4_S and K bands for 2MASS J1404−3159AB that allow us to refine our analysis, and we find the same spectral types as we did in Dupuy & Liu (2012), L9 ± 1 and T5.0 ± 0.5. We have added J-, H-, and K-band flux ratios for 2MASS J2140+1625AB, resulting in the same primary type (M8.0 ± 0.5) but an updated secondary type of L0.5 ± 1.0 (was M9.5 ± 0.5). Finally, for DENIS J2252−1730AB we have new flux ratios in the Y, J, H, CH4_S, and K bands. This allows us to improve the spectral types to L4.0 ± 1.0 (was L4.5 ± 1.5) and T3.5 ± 0.5 (was T3.5 ± 1.0).

Our joint analysis of resolved relative astrometry and unresolved absolute astrometry provides direct measurements of the relative orbit, parallax, and photocenter orbit. The first two of these give the total system mass directly. When the flux ratio of the binary is known, then the photocenter orbit can also be used to directly measure the individual component masses. Our absolute astrometry from CFHT/WIRCam is all in either the J or ${K}_{{\rm{H}}2}$ band, and we have resolved J- and K-band photometry for all of our binaries. Therefore, we can derive individual masses for any binary with a sufficiently well-constrained photocenter orbit semimajor axis (${a}_{\mathrm{phot}}$). As noted above, we assume a uniform prior in the ratio of the photocenter semimajor axis to the true semimajor axis (${a}_{\mathrm{phot}}$/a). If we define the ratio of the secondary-to-total mass as $\mu \equiv {M}_{2}/({M}_{1}+{M}_{2})$ and the ratio of secondary to total flux as $\beta \equiv {F}_{2}/({F}_{1}+{F}_{2})$, then ${a}_{\mathrm{phot}}/a=\mu -\beta$. An unbounded and uniform prior on ${a}_{\mathrm{phot}}$/a and β, as we have assumed, can in principle allow for unphysical values of μ < 0. This is simply a consequence of not using any information about the flux ratio in our astrometric analysis, motivated by the fact that improved flux ratios could easily be obtained in the future. Therefore, not all astrometric solutions result in meaningful constraints on the individual mass ratios, but in our sample of well-determined orbits only Kelu-1AB has an ${a}_{\mathrm{phot}}$ uncertainty sufficiently large to encompass a wide range of unphysical values.

In order to derive flux ratios, we used the values of ΔJ and ΔK reported in Table 9. We use the MKO J-band data directly, while for the narrow-band ${K}_{{\rm{H}}2}$ filter on WIRCam we must derive a correction to be applied to our broadband K measurements. As described in Appendix A.2, we derived third-order polynomial relations between the K-band absolute magnitude and both ${K}_{\mathrm{MKO}}-{K}_{{\rm{H}}2}$ color and ${K}_{2\mathrm{MASS}}-{K}_{{\rm{H}}2}$ color. These relations have an rms scatter of 0.021 mag and 0.025 mag, respectively. When applying the offset from these relations we added these values of the rms scatter in quadrature to the flux ratio uncertainty in any case where our observed ΔK was more than 2× larger than the scatter. In other words, for binaries with flux ratios within ≲0.05 mag of unity, we did not add this uncertainty in quadrature.

Table 10 summarizes all of the photocenter orbit sizes and flux ratios measured in our analysis, including systems for which the dynamical masses are not reliable. Table 11 gives the resulting individual masses we derive from our dynamical total masses and photocenter mass ratios (19 systems) or from other information (three systems). Two of the three binaries in our relative orbit sample that do not have absolute astrometry from CFHT have stellar companions with Hipparcos distances (Gl 417BC and HD 130948BC). We provide the model-derived individual masses in Table 11 for reference, but we do not consider them as part of the following analysis of individual masses. The third binary with an orbit but no CFHT data is Gl 569Bab, and for this system we use the mass ratio of M₁/M₂ = 1.4 ± 0.3 derived from a joint analysis of all available radial velocity data by Konopacky et al. (2010, see their Section 5.4.6). Gl 569Bab and LHS 1070BC (Köhler et al. 2012) are the only two ultracool binary systems that previously had individual mass determinations in the literature not from our own work. Our sample of 38 objects with individual masses therefore increases the sample size by an order of magnitude.

Table 10.  Summary of Key Parameters from Orbit Analysis

System	${M}_{\mathrm{tot}}$	q	e	P	a	${a}_{\mathrm{phot}}$/a	β
	(${M}_{\mathrm{Jup}}$)	≡M2/M1		(days)	(au)		$\equiv {F}_{2}/({F}_{1}+{F}_{2})$ a
LP 349-25AB	${166}_{-7}^{+6}$	${0.941}_{-0.030}^{+0.029}$	${0.0468}_{-0.0018}^{+0.0019}$	2812 ± 5	2.109 ± 0.027	0.056 ± 0.008	0.4286 ± 0.0018
LP 415-20AB	${248}_{-29}^{+26}$	${0.59}_{-0.12}^{+0.10}$	${0.706}_{-0.012}^{+0.011}$	5410 ± 90	3.73 ± 0.12	0.00 ± 0.03	0.370 ± 0.028
SDSS J0423−0414AB	83 ± 3	0.62 ± 0.04	${0.272}_{-0.007}^{+0.008}$	${4491}_{-22}^{+21}$	${2.291}_{-0.028}^{+0.027}$	−0.023 ± 0.006	0.404 ± 0.013
2MASS J0700+3157AB	${141}_{-5}^{+4}$	1.08 ± 0.05	${0.017}_{-0.007}^{+0.005}$	8720 ± 190	4.25 ± 0.08	0.310 ± 0.011	0.208 ± 0.005
LHS 1901AB	${213}_{-10}^{+9}$	${0.87}_{-0.11}^{+0.09}$	0.8304 ± 0.0009	5921 ± 29	3.76 ± 0.06	−0.009 ± 0.028	0.4750 ± 0.0016
2MASS J0746+2000AB	${160.8}_{-1.7}^{+1.8}$	${0.952}_{-0.027}^{+0.026}$	0.4854 ± 0.0003	4652 ± 11	2.919 ± 0.009	0.069 ± 0.007	0.4186 ± 0.0018
2MASS J0850+1057AB	54 ± 8b	⋯	${0.06}_{-0.06}^{+0.05}$ b	${17600}_{-2300}^{+2700}$ b	${4.98}_{-0.33}^{+0.25}$ b	0.04 ± 0.12	0.225 ± 0.021
2MASS J0920+3517AB	187 ± 11	${1.63}_{-0.09}^{+0.08}$	${0.180}_{-0.007}^{+0.006}$	2651 ± 3	2.11 ± 0.04	0.183 ± 0.010	0.437 ± 0.006
SDSS J0926+5847AB	${38}_{-18}^{+10}$ b	⋯	${0.35}_{-0.16}^{+0.22}$ b	${4700}_{-700}^{+500}$ b	${1.80}_{-0.21}^{+0.14}$ b	−0.07 ± 0.08	0.452 ± 0.030
2MASS J1017+1308AB	${156}_{-18}^{+14}$	0.92 ± 0.08	0.158 ± 0.010	6790 ± 80	3.72 ± 0.13	0.026 ± 0.022	0.454 ± 0.004
SDSS J1021−0304AB	${52}_{-7}^{+6}$ b	⋯	0.38 ± 0.07b	${31000}_{-6000}^{+5000}$ b	${7.2}_{-0.8}^{+0.7}$ b	−0.9 ± 0.3	0.5239 ± 0.0028
2MASS J1047+4026AB	${178}_{-12}^{+11}$	0.82 ± 0.06	0.7485 ± 0.0013	${2397}_{-10}^{+11}$	1.94 ± 0.04	0.019 ± 0.015	0.432 ± 0.011
SDSS J1052+4422AB	${90}_{-5}^{+4}$	0.78 ± 0.07	${0.1399}_{-0.0023}^{+0.0022}$	3144 ± 9	1.86 ± 0.03	−0.165 ± 0.008	0.602 ± 0.020
Gl 417BC	${99.2}_{-3.3}^{+3.0}$	⋯	0.105 ± 0.003	${5714}_{-30}^{+29}$	2.851 ± 0.029	⋯	⋯
LHS 2397aAB	${159}_{-8}^{+7}$	${0.706}_{-0.028}^{+0.027}$	0.351 ± 0.003	${5248}_{-17}^{+18}$	3.15 ± 0.05	0.344 ± 0.009	0.0693 ± 0.0025
DENIS J1228−1557AB	${106}_{-19}^{+16}$ b	⋯	${0.089}_{-0.035}^{+0.027}$ b	${18300}_{-2400}^{+1900}$ b	${6.36}_{-0.35}^{+0.29}$ b	0.9 ± 1.1	0.476 ± 0.008
Kelu-1AB	${180}_{-26}^{+22}$ c	⋯	0.709 ± 0.005	9120 ± 70	${4.75}_{-0.22}^{+0.21}$	−0.7 ± 0.9	0.344 ± 0.010
2MASS J1404−3159AB	${120}_{-13}^{+11}$	0.84 ± 0.06	0.825 ± 0.005	6030 ± 80	${3.15}_{-0.11}^{+0.09}$	−0.165 ± 0.016	0.622 ± 0.006
HD 130948BC	${115.4}_{-2.1}^{+2.2}$	⋯	0.1627 ± 0.0017	3656 ± 4	${2.226}_{-0.013}^{+0.014}$	⋯	⋯
Gl 569Bab	138 ± 7	⋯	0.3186 ± 0.0010	865.89 ± 0.20	0.904 ± 0.015	⋯	⋯
SDSS J1534+1615AB	${46}_{-7}^{+6}$ b	⋯	${0.22}_{-0.22}^{+0.21}$ b	${21000}_{-9000}^{+14000}$ b	${5.3}_{-1.6}^{+2.3}$ b	0.07 ± 0.10	0.5411 ± 0.0011
2MASS J1534−2952AB	99 ± 5	${0.95}_{-0.16}^{+0.13}$	${0.0027}_{-0.0027}^{+0.0028}$	${7434}_{-21}^{+18}$	3.40 ± 0.06	0.02 ± 0.04	0.463 ± 0.003
2MASS J1728+3948AB	${140}_{-8}^{+7}$	${0.93}_{-0.13}^{+0.11}$	${0.015}_{-0.015}^{+0.013}$	${14900}_{-400}^{+600}$	${6.09}_{-0.22}^{+0.17}$	0.04 ± 0.03	0.439 ± 0.012
LSPM J1735+2634AB	187 ± 7	${0.868}_{-0.025}^{+0.023}$	0.497 ± 0.004	7910 ± 90	${4.37}_{-0.06}^{+0.07}$	0.077 ± 0.007	0.3872 ± 0.0025
2MASS J1750+4424AB	${190}_{-50}^{+40}$ b	⋯	${0.73}_{-0.07}^{+0.09}$ b	${78000}_{-22000}^{+15000}$ b	${20}_{-4}^{+3}$ b	−0.8 ± 2.1	0.354 ± 0.008
2MASS J1847+5522AB	${270}_{-31}^{+26}$ b	⋯	${0.09}_{-0.09}^{+0.05}$ b	${16500}_{-1200}^{+2000}$ b	${8.0}_{-0.6}^{+0.7}$ b	0.05 ± 0.08	0.439 ± 0.005
SDSS J2052−1609AB	${69}_{-20}^{+14}$ b	⋯	${0.20}_{-0.11}^{+0.09}$ b	${16000}_{-5000}^{+4000}$ b	${5.0}_{-0.6}^{+0.5}$ b	0.06 ± 0.05	0.490 ± 0.013
2MASS J2132+1341AB	${128}_{-8}^{+7}$	${0.88}_{-0.05}^{+0.04}$	${0.315}_{-0.005}^{+0.004}$	3920 ± 60	2.42 ± 0.04	0.155 ± 0.010	0.314 ± 0.008
2MASS J2140+1625AB	${183}_{-17}^{+14}$	${0.60}_{-0.08}^{+0.07}$	0.196 ± 0.007	8930 ± 90	4.71 ± 0.14	0.042 ± 0.025	0.334 ± 0.016
2MASS J2206−2047AB	${188}_{-17}^{+16}$	${0.84}_{-0.10}^{+0.09}$	0.015 ± 0.008	8750 ± 80	${4.69}_{-0.14}^{+0.13}$	−0.033 ± 0.028	0.4885 ± 0.0023
DENIS J2252−1730AB	101 ± 7	${0.70}_{-0.09}^{+0.08}$	0.334 ± 0.009	${3222}_{-10}^{+9}$	1.95 ± 0.04	0.081 ± 0.027	0.330 ± 0.008

Notes.

^aThis is the ratio of the secondary's flux to the total flux in the bandpass used for our CFHT/WIRCam absolute astrometry observations, which was either a broadband J (MKO) or a narrowband filter in the K band centered at 2.122 μm. ^bWe do not use these orbit determinations in our analysis given their questionable quality (see Section 4.1). ^cThis is the total mass of Kelu-1AB using our CFHT parallax, but we do not use it in our analysis given that parallaxes from the literature differ from ours at a level sufficient to result in significantly different physical properties.

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Table 11.  Summary of Properties for Dynamical Mass Sample

	Primary Component				Secondary Component
System	M (${M}_{\mathrm{Jup}}$)	$\mathrm{log}({L}_{\mathrm{bol}}/{L}_{\odot })$	SpT	${T}_{\mathrm{eff}}$ (K)	M (${M}_{\mathrm{Jup}}$)	$\mathrm{log}({L}_{\mathrm{bol}}/{L}_{\odot })$	SpT	${T}_{\mathrm{eff}}$ (K)	Age (Gyr)
LP 349-25AB	85 ± 4	$-{3.075}_{-0.026}^{+0.027}$	M7 ± 1.0	${2729}_{-27}^{+26}$	80 ± 3	−3.198 ± 0.027	M8 ± 1.0	${2629}_{-27}^{+29}$	${0.271}_{-0.029}^{+0.022}$
LP 415-20AB	${156}_{-18}^{+17}$	−3.010 ± 0.030	M6 ± 1.0	⋯	${92}_{-18}^{+16}$	$-{3.260}_{-0.040}^{+0.030}$	M8 ± 0.5	${2640}_{-30}^{+40}$	⋯
SDSS J0423−0414AB	${51.6}_{-2.5}^{+2.3}$	−4.41 ± 0.04	L6.5 ± 1.5	${1430}_{-40}^{+30}$	${31.8}_{-1.6}^{+1.5}$	−4.72 ± 0.07	T2 ± 0.5	1200 ± 40	${0.81}_{-0.09}^{+0.07}$
2MASS J0700+3157AB	68.0 ± 2.6	−3.95 ± 0.04	L3 ± 1.0	1860 ± 40	${73.3}_{-3.0}^{+2.9}$	−4.45 ± 0.04	L6.5 ± 1.5	⋯	${0.76}_{-0.14}^{+0.09}$
LHS 1901AB	113 ± 8	$-{3.005}_{-0.027}^{+0.026}$	M7 ± 1.0	2844 ± 16	99 ± 7	$-{3.046}_{-0.027}^{+0.026}$	M7 ± 1.0	${2820}_{-10}^{+20}$	⋯
2MASS J0746+2000AB	${82.4}_{-1.5}^{+1.4}$	$-{3.596}_{-0.025}^{+0.028}$	L0 ± 0.5	${2318}_{-23}^{+24}$	78.4 ± 1.4	$-{3.777}_{-0.027}^{+0.028}$	L1.5 ± 0.5	${2134}_{-25}^{+26}$	⋯
2MASS J0920+3517AB	71 ± 5	−4.270 ± 0.030	L5.5 ± 1.0	${1621}_{-30}^{+32}$	${116}_{-8}^{+7}$	−4.340 ± 0.030	L9 ± 1.5	⋯	⋯
2MASS J1017+1308AB	${81}_{-11}^{+10}$	−3.78 ± 0.04	L1.5 ± 1.0	2090 ± 50	75 ± 7	−3.84 ± 0.04	L3 ± 1.0	${2040}_{-50}^{+60}$	⋯
2MASS J1047+4026AB	${97}_{-7}^{+6}$	−3.27 ± 0.04	M8 ± 0.5	${2640}_{-40}^{+50}$	80 ± 6	−3.39 ± 0.05	L0 ± 1.0	${2510}_{-50}^{+60}$	⋯
SDSS J1052+4422AB	51 ± 3	−4.51 ± 0.04	L6.5 ± 1.5	${1366}_{-29}^{+25}$	${39.4}_{-2.7}^{+2.6}$	−4.64 ± 0.07	T1.5 ± 1.0	1270 ± 40	${1.04}_{-0.15}^{+0.14}$
Gl 417BCa	${51.5}_{-1.8}^{+1.7}$	−4.132 ± 0.030	L4.5 ± 1.0	${1639}_{-31}^{+29}$	47.7 ± 1.9	−4.220 ± 0.030	L6 ± 1.0	${1560}_{-26}^{+29}$	${0.49}_{-0.04}^{+0.03}$
LHS 2397aAB	93 ± 4	−3.34 ± 0.04	M8 ± 0.5	2560 ± 50	66 ± 4	−4.48 ± 0.04	⋯	1440 ± 40	${2.8}_{-1.5}^{+2.1}$
2MASS J1404−3159AB	65 ± 6	$-{4.52}_{-0.05}^{+0.06}$	L9 ± 1.0	${1400}_{-50}^{+40}$	${55}_{-7}^{+6}$	$-{4.87}_{-0.07}^{+0.08}$	T5 ± 0.5	1190 ± 50	${3.0}_{-1.3}^{+0.8}$
HD 130948BCa	${59.8}_{-2.1}^{+2.0}$	−3.85 ± 0.06	L4 ± 1.0	${1920}_{-60}^{+70}$	${55.6}_{-1.9}^{+2.0}$	−3.96 ± 0.06	L4 ± 1.0	${1800}_{-70}^{+50}$	0.44 ± 0.04
Gl 569Babb	${80}_{-8}^{+9}$	−3.440 ± 0.030	M8.5 ± 0.5	2420 ± 40	${58}_{-9}^{+7}$	−3.670 ± 0.030	M9 ± 0.5	2170 ± 50	${0.44}_{-0.06}^{+0.05}$
2MASS J1534−2952AB	51 ± 5	−4.91 ± 0.07	T4.5 ± 0.5	${1150}_{-50}^{+40}$	48 ± 5	−4.99 ± 0.07	T5 ± 0.5	${1100}_{-50}^{+40}$	${3.0}_{-0.5}^{+0.4}$
2MASS J1728+3948AB	73 ± 7	$-{4.29}_{-0.05}^{+0.04}$	L5 ± 1.0	1600 ± 40	67 ± 5	−4.49 ± 0.04	L7 ± 1.0	1440 ± 40	${3.4}_{-2.1}^{+2.8}$
LSPM J1735+2634AB	100 ± 4	−3.221 ± 0.028	M7.5 ± 0.5	${2677}_{-26}^{+28}$	87 ± 3	$-{3.445}_{-0.028}^{+0.029}$	L0 ± 1.0	2462 ± 30	⋯
2MASS J2132+1341AB	68 ± 4	−4.22 ± 0.05	L4.5 ± 1.5	${1660}_{-40}^{+50}$	60 ± 4	$-{4.50}_{-0.04}^{+0.05}$	L8.5 ± 1.5	${1400}_{-40}^{+30}$	${1.44}_{-0.37}^{+0.26}$
2MASS J2140+1625AB	${114}_{-12}^{+10}$	−3.20 ± 0.04	M8 ± 0.5	2680 ± 40	${69}_{-9}^{+8}$	−3.51 ± 0.04	L0.5 ± 1.0	${2410}_{-50}^{+60}$	⋯
2MASS J2206−2047AB	${102}_{-11}^{+10}$	$-{3.220}_{-0.040}^{+0.030}$	M8 ± 0.5	2670 ± 40	${86}_{-10}^{+8}$	$-{3.260}_{-0.030}^{+0.040}$	M8 ± 0.5	${2650}_{-30}^{+40}$	⋯
DENIS J2252−1730AB	59 ± 5	$-{4.26}_{-0.04}^{+0.05}$	L4 ± 1.0	1590 ± 50	41 ± 4	$-{4.76}_{-0.07}^{+0.08}$	T3.5 ± 0.5	${1210}_{-40}^{+50}$	${1.11}_{-0.22}^{+0.19}$

Notes. The effective temperatures quoted here are derived from the SM08 models for objects cooler than 2100 K and from the Baraffe et al. (2015, hereafter BHAC) models for objects hotter than 2100 K. System ages are only reported in cases where the observations provide a meaningful constraint, and most of the quoted values are from our total-mass analysis using the SM08 models. The exceptions are LP 349-25AB and Gl 569Bab (total-mass analysis using the BHAC models), 2MASS J0700+3157AB (individual-mass analysis of the primary using SM08 models), and LHS 2397aAB (individual-mass analysis of the secondary using SM08 models). The individual masses given here are directly measured in this work unless otherwise noted.

^aThe individual masses quoted for Gl 417BC and HD 130948BC are not measured directly but rather are inferred from evolutionary models in our total-mass analysis. The component effective temperatures are also from our total-mass analysis. ^bThe individual masses quoted for Gl 569Bab are computed from our total mass and the mass ratio from Konopacky et al. (2010). The component effective temperatures are from our total-mass analysis.

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Figure 3 shows all of our individual masses, as well as those from the literature, as a function of spectral type. In the analysis that follows, we will use this information to discuss the substellar boundary. Here we simply note that in most cases, our individual masses do not grossly deviate from rough astrophysical expectations, e.g., nearly equal flux systems have mass ratios near unity and vice versa. There are a few remarkable exceptions, such as the secondary 2MASS J0920+3517B (L9.0 ± 1.5) that has a mass of ${116}_{-8}^{+7}$ ${M}_{\mathrm{Jup}}$, well in excess of the expected hydrogen fusion limit (≈70 ${M}_{\mathrm{Jup}}$; Section 7.1).

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Figure 4 shows our dynamical mass sample on the color–magnitude diagram (CMD) with data points colored by the measured masses. Overall, mass broadly decreases through the CMD sequence, although there is no one-to-one correspondence between mass and CMD location, as expected for our sample that is drawn from the field population of ultracool dwarfs spanning a wide range of ages. The clearest illustration of this is that the lowest mass objects happen to be located roughly in the middle of the L/T transition, while the latest-type, bluest objects are somewhat more massive. This more likely reflects the distribution of ages in our sample rather than indicating that later-type objects tend to be more massive. Figure 5 shows our measured absolute magnitudes as a function of spectral type, colored according to mass, in order to assess the reliability of both plotted quantities. Our sample is broadly consistent with the field sequence, indicating the accuracy of our spectral types, with the largest outlier being 2MASS J0920+3517B, which is consistent with our dynamical masses showing that this is an unresolved pair of brown dwarfs in a triple system.

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To provide a simple summary of our results, Table 12 reports the mean mass in seven spectral type bins selected to contain three to seven objects per bin, excluding unusual objects (unresolved binaries and the pre-main-sequence system LP 349-25AB). As expected, the mean mass broadly decreases with spectral type. We caution that this table is simply meant to provide a guide to the typical masses of objects in the field population and is not intended to be used for any quantitative astrophysical purpose. At earlier spectral types, Table 12 may be useful for providing model-independent mass estimates for very low-mass stars. Next, we briefly compare these results to previous work and defer discussion of the astrophysical interpretation of our individual masses to later sections.

Table 12.  Average Mass Per Spectral Type Bin

Spectral	Mass (${M}_{\mathrm{Jup}}$)		Number
Type	mean	rms	in bin
M7–M7.5	102	8	3
M8–M8.5	94	11	7
M9–L0.5	81	10	7
L1–L3.5	76	6	4
L4–L7.5	58	9	7
L8–T1.5	48	14	3
T2–T5.5	36	9	5

Note. The mean is computed as the weighted average among the individual mass measurements in each spectral type bin. Bin sizes vary in order to include a reasonable number of objects per bin, and the bins are larger at later types. From our own 38 individual masses we exclude the pre-main-sequence binary LP 349-25AB, objects that are possible unresolved binaries (LP 415-20A, 2MASS J0700+3157B, and 2MASS J0920+3517B), and LHS 2397aB (66 ± 4 ${M}_{\mathrm{Jup}}$) because it has no directly measured spectral type. Individual masses from the literature that we include here are those for LHS 1070BC (M9.5+L0; Köhler et al. 2012) and Gl 569Bab (M8.5+M9; total mass from this work and mass ratio from Konopacky et al. 2010).

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5.2.1. Comparison to Literature Mass Ratios

We have derived luminosities for all objects in our sample using our resolved photometry along with the empirical relations between luminosity and absolute magnitude that are described in Appendix A.3. This is somewhat different from our past work where we relied on bolometric correction–spectral type relations to derive individual luminosities. Our new approach obviates the need to reference spectral types, as it links absolute magnitude directly to luminosity. Given that J- and H-band absolute magnitudes get brighter and plateau across the L/T transition, respectively, only K-band absolute magnitudes are suitable for the entire range of luminosity of our sample. The luminosity scatter about the K-band polynomial relation is somewhat higher across the L/T transition (0.07 dex; M_K > 13.0 mag) than at brighter magnitudes (0.04 dex). Interestingly, if we only consider the late-M and earlier L dwarfs with M_H < 13.3 mag, the scatter in luminosity about the H-band relation is significantly lower (0.023 dex). It is not obvious why this should be the case from an astrophysical perspective, but we nonetheless use this to our advantage. For binaries where both components have M_H < 13.3 mag, we use the H-band relation with luminosity, and for the remainder we use the K-band relation. In all cases we used whichever photometry was more precise between 2MASS and MKO. We report the resulting component luminosities in Table 11, and Figure 6 shows our derived luminosities as a function of our individual masses.

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As a check on our method, we compare our results to published work using integrated-light spectra and photometry to derive luminosities. For objects in common with Golimowski et al. (2004a), Dieterich et al. (2014), and Filippazzo et al. (2015), all of whom use somewhat different methods and generations of models for deriving luminosities, we find that our total luminosities agree well within the errors after accounting for the different distances assumed in each work. For example, summing our luminosities for the components of 2MASS J0746+2000AB, we find $\mathrm{log}({L}_{\mathrm{bol}}/{L}_{\odot })=-3.375\pm 0.020$ dex, which agrees well with published values of −3.41 ± 0.02 dex (Golimowski et al. 2004a), −3.413 ± 0.009 dex (Dieterich et al. 2014), and −3.391 ± 0.003 dex (Filippazzo et al. 2015). Other objects in common include Kelu-1, LHS 2397a, SDSS J0423−0414, and 2MASS J1728+3948.

Our work represents the first sample of individual masses for spectrally classified L and T dwarfs. This allows us to examine the maximum mass of the latest-type objects, which is the mass of the boundary between stars and brown dwarfs. Figure 6 shows our 38 individual mass and luminosity measurements along with all other published model-independent masses, including the spectrally unclassified objects Gl 802B (80 ± 14 ${M}_{\mathrm{Jup}}$; Ireland et al. 2008) and HR 7672B (${68.7}_{-3.1}^{+2.4}$ ${M}_{\mathrm{Jup}}$; Crepp et al. 2012), as well as the M8.5+M9 binary Gl 569Bab (${80}_{-8}^{+9}$ ${M}_{\mathrm{Jup}}$ and ${58}_{-9}^{+7}$ ${M}_{\mathrm{Jup}}$; Konopacky et al. 2010), the M9.5+L0 binary LHS 1070BC (81 ± 5 ${M}_{\mathrm{Jup}}$ and 74 ± 4 ${M}_{\mathrm{Jup}}$; Köhler et al. 2012), and the M7+M7 binary LSPM J1314+1320AB (92.8 ± 0.6 ${M}_{\mathrm{Jup}}$ and 91.7 ± 1.0 ${M}_{\mathrm{Jup}}$; Dupuy et al. 2016). None of the components violate the most basic predictions of models within the observational uncertainties, except for systems suspected to be higher-order multiples based on our analysis independent of models. No objects have a lower luminosity than expected given their mass and the finite age of the universe.

Figure 3 shows our individual masses as a function of spectral type. The lack of objects at high mass and late spectral type is empirical evidence for a mass limit to hydrogen fusion. Moreover, examining the maximum mass of the latest-type objects (or equivalently the lowest luminosity objects in Figure 6) allows us to constrain the upper limit on the mass of the substellar boundary, and correspondingly a lower limit on the mass of main-sequence stars. In fact, our dynamical mass sample is (somewhat unfortunately) ideal for this test because we are biased toward high masses. The main observational limitation in measuring masses to date has been achieving the time baseline needed for robust orbit determinations of binaries with typical periods of 20–30 years. The highest mass late-L or T dwarfs in our sample are 2MASS J1728+3948B (L7.0 ± 1.0; 67 ±5 ${M}_{\mathrm{Jup}}$; $\mathrm{log}({L}_{\mathrm{bol}}/{L}_{\odot })=-4.49\pm 0.04$ dex) and 2MASS J1404−3159A (L9.0 ± 1.0; 65 ± 6 ${M}_{\mathrm{Jup}}$; $\mathrm{log}({L}_{\mathrm{bol}}/{L}_{\odot })=-4.52\,\pm 0.05$ dex). The next most massive late-type object is 2MASS J2132+1341B (L8.5 ± 1.5; 60 ± 4 ${M}_{\mathrm{Jup}}$; $\mathrm{log}({L}_{\mathrm{bol}}\,/{L}_{\odot })=-{4.50}_{-0.04}^{+0.05}$ dex). Going to earlier types, there are three objects that have masses and spectral types that are consistent among each other within the errors: 2MASS J0920+3517A (L5.5 ± 1.0; 71 ± 5 ${M}_{\mathrm{Jup}}$; $\mathrm{log}({L}_{\mathrm{bol}}/{L}_{\odot })=-4.28\pm 0.03$ dex), 2MASS J1728+3948A (L5.0 ± 1.0; 73 ± 7 ${M}_{\mathrm{Jup}}$; $\mathrm{log}({L}_{\mathrm{bol}}\,/{L}_{\odot })=-{4.29}_{-0.05}^{+0.04}$ dex), and 2MASS J2132+1341A (L4.5 ± 1.5; 68 ± 4 ${M}_{\mathrm{Jup}}$; $\mathrm{log}({L}_{\mathrm{bol}}/{L}_{\odot })=-4.22\pm 0.05$ dex). Despite the fact that we are more sensitive to massive systems, even our most massive late-type objects are only consistent with having masses as high as 71 ${M}_{\mathrm{Jup}}$ within 1σ, and even at spectral types as early as L4, the most massive objects are consistent with 70 ${M}_{\mathrm{Jup}}$. Therefore, we estimate an empirical substellar boundary of ≈70 ${M}_{\mathrm{Jup}}$.

The theoretical mass limit of hydrogen fusion quoted in the literature varies widely. One review by Burrows et al. (2001) places the minimum mass for H fusion between 0.070–0.075 M_⊙ (73–79 ${M}_{\mathrm{Jup}}$) for solar metallicity and at 0.092 M_⊙ (96 ${M}_{\mathrm{Jup}}$) for zero metallicity. More recent models from Burrows et al. (2011) also show a range of 0.070–0.075 M_⊙ for the H-fusion mass assuming different helium fractions (their Figure 6). A review by Chabrier & Baraffe (2000) gives the minimum mass for H fusion as 0.070 M_⊙ (73 ${M}_{\mathrm{Jup}}$) when cloud opacity is included and 0.072 M_⊙ (75 ${M}_{\mathrm{Jup}}$) for cloudless atmospheres. If the H-fusion mass limit is indeed as high as ≈75 ${M}_{\mathrm{Jup}}$, then it is surprising that we have not uncovered objects with masses closer to this limit in our sample. Figure 7 shows more recent tracks from BHAC that includes a 0.072 M_⊙ (75 ${M}_{\mathrm{Jup}}$) track that stabilizes in luminosity after ≈2 Gyr and thus we would consider a star. Some of the next lower mass tracks appear more star-like in that sense, but by 0.067 M_⊙ (70 ${M}_{\mathrm{Jup}}$) the luminosity appears to be on course to monotonically decrease with the age of the universe. Finally, SM08 quote a minimum mass of 0.070 M_⊙ (73 ${M}_{\mathrm{Jup}}$) from their cloudy models, because that track ultimately reaches 1550 K where clouds are very significant in the atmosphere. Thus, the lack of any ≳70 ${M}_{\mathrm{Jup}}$ brown dwarfs in our sample is consistent with the lower range of predictions in the literature for the minimum mass for H fusion (70–73 ${M}_{\mathrm{Jup}}$).

If we adopt our empirical substellar boundary of ≈70 ${M}_{\mathrm{Jup}}$, we can then determine the latest spectral type of objects above this boundary (i.e., objects that may be stars at the very bottom of the main sequence). The most precise stellar mass in our sample is for 2MASS J0746+2000B (L1.5 ± 0.5; 78.4 ± 1.4 ${M}_{\mathrm{Jup}}$; $\mathrm{log}({L}_{\mathrm{bol}}/{L}_{\odot })=-{3.777}_{-0.027}^{+0.028}$ dex), but this is likely well above the H-fusion limit. Going to later spectral types we find 2MASS J0700+3157A (L3.0 ± 1.0; 68.0 ± 2.6 ${M}_{\mathrm{Jup}}$; $\mathrm{log}({L}_{\mathrm{bol}}/{L}_{\odot })=-3.95\pm 0.04$ dex) and 2MASS J1017+1308B (L3.0 ± 1.0; 75 ± 7 ${M}_{\mathrm{Jup}}$; $\mathrm{log}({L}_{\mathrm{bol}}/{L}_{\odot })=-3.84\pm 0.04$ dex). After this is the cluster of ≈L4–L6 objects with masses consistent with 70 ${M}_{\mathrm{Jup}}$ mentioned above. We therefore conclude that the end of the main sequence occurs within the range of spectral types L3–L5 and luminosities of 10^−4.3 to 10^−3.9 dex, independent of model assumptions. Objects with spectral types later than this, or luminosities lower than this, are brown dwarfs, while objects with earlier types or higher luminosities could either be stars or brown dwarfs. Our result is consistent with early works on the first samples of L dwarfs that showed the substellar boundary was likely within this spectral type range (e.g., Kirkpatrick et al. 1999). A larger sample and more precise component spectral types would enable us to refine the location of this empirically defined substellar boundary.

Finally, we discuss our results in the context of the work of Dieterich et al. (2014) on the substellar boundary. Their approach uses the minimum radius (determined via luminosity and a temperature derived from model atmospheres) on the Hertzsprung–Russell diagram. The physical motivation is that the transition to degeneracy-dominated interiors marks the beginning of the substellar regime, as degeneracy pressure is what prevents objects below a critical mass from reaching sufficiently high core temperatures to generate a significant amount of energy by fusing hydrogen. To perform their test, they start at the late-M dwarfs and examine progressively cooler objects that have progressively smaller radii, until the trend reverses and cooler objects no longer have significantly smaller radii. Dieterich et al. (2014) find that this occurs at a radius of 0.086 ± 0.003 ${R}_{\odot }$ for the object 2MASS J0523−1403, which has a spectral type of L2.5 and $\mathrm{log}(L/{L}_{\odot })=-3.9$ dex, and they conclude that this marks the end of the main sequence. Compared to the boundary derived here using mass, their result is somewhat earlier in type and more luminous than our result but broadly consistent within the uncertainties. Dieterich et al. (2014) discuss the fact that models predict a locus for their radius test at somewhat cooler temperatures and suggest that lowering the metal abundances adopted in current models could explain the discrepancy; however, they also note that such a change in the abundances would alter the mass limit of the boundary to higher masses. The fact that we find a rather low mass limit for the substellar boundary would therefore seem to indicate that abundances alone will not rectify the tensions between models and the results of Dieterich et al. (2014). We speculate that one possibility could be systematic errors in temperatures derived from model atmospheres akin to those identified in our previous work on late-M dwarfs (Dupuy et al. 2010), where model atmospheres gave higher temperatures than implied by evolutionary model radii and measured luminosities.

Theoretical predictions of lithium depletion in ultracool dwarfs as a function of mass and age are thought to be one of the most reliable methods for age-dating young clusters (e.g., Chabrier et al. 1996; Bildsten et al. 1997; Binks & Jeffries 2014; Kraus et al. 2014). They have also long been used as a method of confirming ultracool dwarfs as substellar (e.g., Rebolo et al. 1992; Basri et al. 1996). The brown dwarfs in our sample span masses both above and below the predicted lithium depletion boundary, and many of them are in systems with published optical, integrated-light spectroscopy that constrains the presence or absence of lithium. One potential complication is that the Li i doublet at 6708 Å, which provides the most sensitive observational probe of lithium, is expected to disappear in cool objects where monatomic lithium becomes incorporated into LiCl and LiOH molecules or at cooler temperatures condenses into LiF and Li₂S (e.g., Lodders 1999). This chemical depletion at low temperatures has not previously been tested with objects of known mass. The latest-type object known to display Li i absorption is WISE J1049−5319B (T0; Faherty et al. 2014; Lodieu et al. 2015), implying that any of the L dwarfs in our sample might plausibly display Li i absorption if they are not depleted by Li fusion.

Figure 9 shows our individual masses as a function of age for the 13 systems with well-constrained model-derived ages, i.e., systems containing at least one brown dwarf, as well as the pre-main-sequence binary LP 349–25AB. (The low-mass main-sequence binaries in our sample have unconstrained ages, and as expected, none are known to display Li i absorption.) Only 2 of these 13 binaries have published lithium detections, SDSS J0423−0414AB (EW = 11 Å; Kirkpatrick et al. 2008) and Gl 417BC (EW = 11.5 Å; Kirkpatrick et al. 2001). Seven more systems have published spectra that exclude lithium absorption in integrated light: LP 349–25AB (<0.5 Å; Reiners & Basri 2009), 2MASS J0700+3157AB (<0.3 Å; Thorstensen & Kirkpatrick 2003), LHS 2397aAB (<0.5 Å; Reiners & Basri 2009), 2MASS J1728+3948AB (<4 Å; Kirkpatrick et al. 2000), 2MASS J2132+1341AB (Cruz et al. 2007), and DENIS J2252−1730AB (Reid et al. 2008b). (See Appendix B for a detailed discussion of all of these binaries and their lithium constraints.)

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Also shown in Figure 9 are the lithium-depletion mass limits predicted by BHAC models and by the Tucson models (Burrows et al. 1997). Although we do not use the Tucson models in our other model analysis, mainly because in addition to their non-gray atmosphere models they used gray atmospheres at the higher temperatures (including some temperatures relevant for our sample) in order to cover the full range of evolution, we include them here because they are the only other available evolutionary tracks that report lithium depletion. As expected, lithium nondetections in the most massive objects, LHS 2397aA (M8, 93 ± 4 ${M}_{\mathrm{Jup}}$) and the young (${271}_{-29}^{+22}$ Myr, BHAC) components of LP 349-25AB (M7+M8, 85 ± 4 ${M}_{\mathrm{Jup}}$ and 80 ± 3 ${M}_{\mathrm{Jup}}$), are all consistent with model-predicted lithium depletion. Likewise, the two systems with strong lithium detections comprise some of the lowest mass components in the sample (≲50 ${M}_{\mathrm{Jup}}$), so they are consistent with both sets of models that predict they should be abundant in lithium.

The lowest mass objects with lithium not detected are the 59 ± 5 ${M}_{\mathrm{Jup}}$ and 41 ± 4 ${M}_{\mathrm{Jup}}$ components of DENIS J2252−1730AB (L4+T3.5; ${1.10}_{-0.18}^{+0.15}$ Gyr). Within the 1σ uncertainties, the primary is consistent with being strongly depleted according to the BHAC models, $\mathrm{log}(\mathrm{Li}/{\mathrm{Li}}_{\mathrm{init}})=-{2.1}_{-1.5}^{+0.7}$ dex (Cond), and the late-type secondary is likely too cool to possess monatomic lithium. In contrast, the Tucson models predict that DENIS J2252−1730A should have retained most of its lithium, $\mathrm{log}(\mathrm{Li}/{\mathrm{Li}}_{\mathrm{init}})=-{0.04}_{-0.06}^{+0.04}$ dex, and are only consistent with full depletion at 2.0σ. Five other objects in four systems with masses below 70 ${M}_{\mathrm{Jup}}$ (2MASS J2132+1341AB, 2MASS J0700+3157A, LHS 2397aB, 2MASS J1728+3948B) are all consistent at ≤1σ with full lithium depletion according to the BHAC models. In contrast, Tucson models predict that all of these objects could possess a substantial fraction (up to 100%) of their initial lithium within the 1σ errors. This can be seen visually in Figure 9 as the cluster of lithium nondetections around 60–70 ${M}_{\mathrm{Jup}}$ and 2 Gyr that mostly lie above the BHAC 99.9% depletion curve but that lie entirely below the Tucson 99.9% depletion curve.

We have noted this tension between the predictions of the Lyon and Tucson models in our previous work on HD 130948BC (Dupuy et al. 2009b). Our large sample of individual masses below the substellar boundary suggests that lithium is destroyed via fusion at lower masses than the Tucson models predict but at masses consistent with the BHAC models. The fact that the Tucson models predict a higher mass limit for lithium fusion indicates that their central temperatures are lower, which can be understood as a consequence of their atmospheres being less opaque as discussed in detail in Section 3.2 of SM08. According to the BHAC models, at field ages of ≳1 Gyr, the lithium depletion boundary is at a mass of 60 ${M}_{\mathrm{Jup}}$. Some of the most interesting systems for testing lithium depletion are the young (400–800 Myr) binaries Gl 569Bab, HD 130948BC, and Gl 417BC, which span masses of ≈50–70 ${M}_{\mathrm{Jup}}$ but do not yet have directly measured individual masses and/or optical spectra. Individual masses are attainable in the future with absolute astrometry, either relative to their stellar hosts (Gl 569Bab, HD 130948BC) or from wide-field HST imaging (Gl 417BC). In addition, resolved optical spectra from HST/STIS would benefit the entire sample, enabling stricter tests of models than are currently possible with integrated-light spectra.

In principle, every binary in our sample can be thought of as a "mini cluster" of coeval, co-compositional objects that can be used to test model isochrones (e.g., Liu et al. 2010). The most straightforward approach is to frame this as a test of coevality. Given our directly measured individual masses and luminosities, we derive from models an age posterior distribution for each component (t₁ and t₂). We then examine the posterior distribution of the difference in ages, and the extent to which it is different from zero corresponds to a discrepancy in the mass–luminosity relation predicted by model isochrones. Because we use a Monte Carlo approach, covariances due to parameters in common between components (distance and total mass) are naturally accounted for in our analysis.

Figure 10 displays the median and 1σ intervals of the posterior distributions for ${\rm{\Delta }}\mathrm{log}(t)\equiv \mathrm{log}{t}_{2}-\mathrm{log}{t}_{1}$. To examine how this coevality criterion depends on the underlying physical properties, we plot these values as a function of the secondary component mass (i.e., the component that is less luminous). At the highest masses, where the binaries are composed of two main-sequence stars that each have unconstrained age posteriors, the resulting coevality values are all ${\rm{\Delta }}\mathrm{log}t\approx 0.0\pm 0.5$ dex at 1σ. In other words, both stars are consistent with the full range of main-sequence ages (a few hundred Myr, depending on mass, to 10 Gyr, the maximum age of the models).

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The more interesting coevality tests are for binaries composed of two brown dwarfs. All but two of these systems give consistent ages for the primary and secondary at ≲1σ across all models. The exceptions are two binaries spanning the L/T transition that have the lowest-mass secondaries of any objects in our sample. According to the Lyon Cond models, SDSS J0423−0414AB and SDSS J1052+4422AB are 3.0σ and 1.5σ discrepant with coevality, respectively. In contrast, SM08 models predict a shallower mass–luminosity relation in the L/T transition and thus yield coeval ages for both systems (consistent at 0.5σ and 0.2σ, respectively). The key difference between the Lyon models (Cond and Dusty) and the SM08 hybrid models is that SM08 prescribes an ad hoc change from a cloudy to a cloudless photosphere as the temperature drops from 1400 K to 1200 K. This results in luminosity dropping less quickly during and immediately following the transition from cloudy to cloudless atmospheres.

We originally discussed this discrepancy between the SM08 and Lyon models for SDSS J1052+4422AB in Dupuy et al. (2015b). The case of SDSS J0423−0414AB (L6.5 ± 1.5 and T2.0 ± 0.5) is very similar to SDSS J1052+4422AB (L6.5 ± 1.5 and T1.5 ± 1.0). The primary masses are the same within the errors (${51.6}_{-2.5}^{+2.3}$ ${M}_{\mathrm{Jup}}$ and 51 ± 3 ${M}_{\mathrm{Jup}}$, respectively), but SDSS J0423−0414AB has a lower mass ratio (0.62 ± 0.04 compared to 0.78 ± 0.07) and luminosity ratio (${\rm{\Delta }}\mathrm{log}({L}_{\mathrm{bol}})={0.31}_{-0.08}^{+0.09}$ dex compared to 0.13 ± 0.08 dex). Thus, SDSS J0423−0414AB has even more leverage in our coevality test, and indeed it confirms our previous findings at even higher significance. The reason for the coevality test failure in both cases is that the secondary is more luminous (or equivalently the primary is less luminous) than predicted for their masses at a single age in Lyon models. According to the SM08 hybrid models, luminosity does not fade as quickly during and immediately following the transition from cloudy to cloud-free atmospheres. So according to the SM08 models, it is the secondary that is more luminous than expected (rather than a suppressed luminosity of the primary) because the secondary has already cooled through the L/T transition and lost its clouds.

A new physical explanation for the L/T transition has been proposed by Tremblin et al. (2016), where the primary driver is a thermo-chemical instability in the carbon chemistry (CO/CH₄). Our measurement of the mass–luminosity relation through the L/T transition will provide a key test of this new idea after it is implemented in evolutionary models in the future.

Our individual mass and luminosity measurements have revealed that some objects in our sample are likely triple systems where the higher-order multiplicity is unresolved by current observations. The most extreme case is 2MASS J0920+3517AB (L5.5+L9), where the total mass of 187 ± 11 ${M}_{\mathrm{Jup}}$ is high enough that it suggests the presence of an unresolved massive component. Our individual masses reveal that indeed the less luminous (by 0.06 ± 0.04 dex) component 2MASSJ0920+3517B is much more massive (${116}_{-8}^{+7}$ ${M}_{\mathrm{Jup}}$) than the more luminous 2MASS J0920+3517A (71 ± 5 ${M}_{\mathrm{Jup}}$). In fact, the anomalous mass of 2MASS J0920+3517B would be obvious on its own, given that its spectral type is L9.0 ± 1.5, and that models cannot reproduce its luminosity at such a high mass (Figure 11). 2MASS J0920+3517B can plausibly be explained by a simplistic model in which it is composed of two equal-luminosity components with equal masses of ${58}_{-4}^{+3}$ ${M}_{\mathrm{Jup}}$. The unresolved pair must be rather tight, as we have never observed an elongated PSF for the secondary relative to the primary, and the astrometric residuals about the resolved orbit fit have an rms of 0.5 mas for the better half (13 out of 25) of our measurements. This implies an inner orbit that is much smaller than the observed outer orbit (a = 68.15 ± 0.23 mas, 2.11 ± 0.04 au). We discuss this system in more detail in Appendix B.7.

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View extended figure (3 panels)

Unlike 2MASS J0920+3517AB, the total mass of the L3+L6.5 binary 2MASS J0700+3157AB (${141}_{-5}^{+4}$ ${M}_{\mathrm{Jup}}$) is not unusually high. Indeed, our total-mass model analysis readily finds a self-consistent way to apportion the mass of the two components (${76.8}_{-1.3}^{+1.5}$ ${M}_{\mathrm{Jup}}$ and ${66}_{-3}^{+4}$ ${M}_{\mathrm{Jup}}$, Cond) according to their quite different luminosities (${\rm{\Delta }}\mathrm{log}({L}_{\mathrm{bol}}/{L}_{\odot })\,=0.50\,\pm 0.06$ dex) at an age of ${2.1}_{-0.6}^{+0.4}$ Gyr. However, our directly measured masses are 68.0 ± 2.6 ${M}_{\mathrm{Jup}}$ and ${73.3}_{-3.0}^{+2.9}$ ${M}_{\mathrm{Jup}}$. The less luminous component is again the more massive one, implying that 2MASS J0700+3157B is in fact an unresolved binary. The model-derived ages for 2MASS J0700+3157A are ${0.76}_{-0.14}^{+0.09}$ Gyr (SM08) and ${1.01}_{-0.19}^{+0.13}$ Gyr (Cond). If we assume that 2MASS J0700+3157B is composed of equal-luminosity components with equal masses of ${36.7}_{-1.5}^{+1.4}$ ${M}_{\mathrm{Jup}}$, their model-derived ages are somewhat inconsistent with the primary's (${1.15}_{-0.12}^{+0.11}$ Gyr and ${0.83}_{-0.09}^{+0.08}$ Gyr, respectively), suggesting that this simplistic scenario is not likely. As with 2MASS J0920+3517AB, we have never observed evidence for PSF elongation of 2MASS J0700+3157B in our Keck LGS AO imaging, and the rms about the relative orbit fit is 0.22 mas for the better half (11 out of 22) of our measurements. Thus, the inner orbit in this system is likely very small relative to the outer orbit ($a={377}_{-6}^{+5}$ mas, 4.25 ± 0.08 au). We discuss this system in more detail in Appendix B.4.

We have identified a third candidate triple system. The total mass of LP 415-20AB (${248}_{-29}^{+26}$ ${M}_{\mathrm{Jup}}$) is very high for such late spectral type components (M6+M8). Our individual masses show that this is because the primary is very massive (${156}_{-18}^{+17}$ ${M}_{\mathrm{Jup}}$), while the secondary's mass (${92}_{-18}^{+16}$ ${M}_{\mathrm{Jup}}$) is consistent with its luminosity according to the BHAC models. Given the relatively large individual mass uncertainties, it is unclear whether the putative unresolved component of LP 415-20A is another low-mass star or if it is a brown dwarf. It is less likely, but still possible, that the apparent discrepancy here is due to the rather large observational uncertainties. An improved parallax for this system, which is also notable as a possible member of the Hyades, will help clarify the situation as we discuss in more detail in Appendix B.2.

2MASS J0700+3157 and 2MASS J0920+3517 bring the tally of high-order multiples composed entirely of likely brown dwarfs to five, and these are the first such triple systems with directly measured masses, semimajor axes, and eccentricities. The first triple brown dwarf identified was DENIS J020529.0−115925 (Bouy et al. 2005), where the primary (L5) is orbited by an L/T transition binary (L8+T0). Radigan et al. (2013) discovered the triple T dwarf system 2MASS J08381155+1511155, where the brightest component (T3) is orbited by two slightly fainter components (T3+T4.5). Stone et al. (2016) discovered that the primary in the young system VHS J125601.92−125723.9 (Gauza et al. 2015) is actually a binary, making this the only one of these hierarchical triples in which the inner pair contains the most massive components. (Note that the distance of VHS J1256−1257 is not secure, and the more massive inner pair could be composed of low-mass stars and not brown dwarfs.) The hierarchical mass ordering of the DENIS J0205−1159 and 2MASS J0838+1511 systems appear to be similar to 2MASS J0700+3157 and 2MASSJ0920+3517, but the relative sizes of the inner and outer orbits may be different. The inner and outer pairs of DENIS J0205−1159 have projected separations of 1.3 au and 7 au, respectively, and 2MASS J0838+1511 has projected separations of 2.5 au and 27 au. If the inner pairs of 2MASS J0700+3157 and 2MASS J0920+3517 were only a factor of 7 × –10× smaller than the orbit, then we would have either resolved them directly or possibly seen perturbations in our relative astrometry. Thus, it is likely that the semimajor axis ratios of 2MASS J0700+3157 and 2MASS J0920+3517 are much smaller than for DENIS J0205−1159 and 2MASS J0838+1511. Finally, we note that the eccentricity of the outer orbit is very low for 2MASS J0700+3157 (${0.017}_{-0.007}^{+0.005}$) and is also rather low for 2MASS J0920+3517 (${0.180}_{-0.007}^{+0.006}$). This would seem to rule out a violent dynamical origin for these triple systems.

Without directly measured radii for field brown dwarfs of known spectral type, all previous works on spectral type–${T}_{\mathrm{eff}}$ relations have relied on assumptions about age combined with model radii (e.g., Golimowski et al. 2004a; Vrba et al. 2004; Stephens et al. 2009; Filippazzo et al. 2015). Our sample of visual binaries with directly measured luminosities and precise model-derived ages and radii allows us to examine spectral type–${T}_{\mathrm{eff}}$ relations without uncertainties in radius, due to the unknown ages or masses. We consider the effective temperatures derived from the BHAC and Cond models as a single "Lyon" relation, using BHAC results when available. We used results from our individual-mass analysis when possible, relying on our total-mass analysis for the three systems without astrometric mass ratios (Gl 417BC, Gl 569Bab, and HD 130948BC). We excluded the three likely unresolved binaries LP 415-20A, 2MASS J0700+3157B, and 2MASS J0920+3517B. LHS 2397aB is also excluded because it lacks a spectral type determination. This results in a sample of 40 objects with Lyon temperatures and 22 objects with SM08 temperatures. (SM08 results are only for the coolest objects; our highest SM08 model-derived ${T}_{\mathrm{eff}}$ is 2090 ± 50 K for 2MASS J1017+1308A.)

Figure 12 shows our model-derived effective temperatures as a function of spectral type. We calculated second-order polynomial fits of ${T}_{\mathrm{eff}}$ as a function of spectral type, weighting by the quadrature average of the +1σ and −1σ uncertainties in ${T}_{\mathrm{eff}}$. The coefficients of these fits are given in Table 14. The Lyon fit covers spectral types of M7–T5 and ${T}_{\mathrm{eff}}\approx 1100\mbox{--}2800$ K, and the residuals have an rms scatter of 90 K. The SM08 fit covers spectral types of L1.5–T5 and ${T}_{\mathrm{eff}}\approx 1100\mbox{--}2100$ K, and the residuals have an rms scatter of 80 K. The two fits agree reasonably well in the overlapping spectral type range, with the main difference being that the SM08 fit gives ≈100 K cooler temperatures for L4–L7 dwarfs. This seems to be a reflection of the fact that the SM08 model-derived temperatures for these objects are indeed systematically lower, due to SM08 radii being ≈10% larger than Cond radii at these temperatures, and not simply a quirk of the polynomial fit. This is expected from the fact that the SM08 models have clouds while Cond models do not (e.g., see the discussion of cloudy versus clear models by Burrows et al. 2011). There are no significant outliers in the SM08 fit, and in the Lyon fit only Gl 569Bb appears to be ≳1.5σ discrepant. It has a model-derived temperature that is 310 K cooler than expected given its M9 spectral type and more consistent with L0.5–L1 types. Gl 569Ba is also slightly discrepant for its M8.5 spectral type (140 K cooler than the fit). Therefore, Gl 569Bb may appear discrepant because the system as a whole is a slight outlier in ${T}_{\mathrm{eff}}$ due to the shared distance uncertainty in addition to the spectral type of Gl 569Bb perhaps being slightly underestimated.

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Table 14.  Coefficients of Polynomial Fits

y	x	c0	c1	c2	c3	c4	c5	rms	Valid x Range
${T}_{\mathrm{eff}}$ (Lyon)	SpT	4251.0	−238.03	4.582	⋯	⋯	⋯	90 K	M7–T5
${T}_{\mathrm{eff}}$ (SM08)	SpT	4544.3	−284.52	6.001	⋯	⋯	⋯	80 K	L1.5–T5
MKO Photometric System
$\mathrm{log}({L}_{\mathrm{bol}}/{L}_{\odot })$	MK	−172.188	68.11147	−10.671188	0.8162709	−0.03068824	0.000454547	0.05 dexa	9.1–17.8 mag
$\mathrm{log}({L}_{\mathrm{bol}}/{L}_{\odot })$	MH	−13.282	3.46876	−0.351721	0.0106200	⋯	⋯	0.023 dex	9.6–13.3 mag
${J}_{\mathrm{MKO}}-{J}_{2\mathrm{MASS}}$	MK	−2.284	0.55165	−0.043190	0.0010379	⋯	⋯	0.015 mag	9.1–17.0 mag
${H}_{\mathrm{MKO}}-{H}_{2\mathrm{MASS}}$	MH	−0.051	0.00850	⋯	⋯	⋯	⋯	0.003 mag	9.6–13.4 mag
${H}_{\mathrm{MKO}}-{H}_{2\mathrm{MASS}}$	MK	−0.702	0.15644	−0.010605	0.0002365	⋯	⋯	0.004 mag	9.1–17.0 mag
${K}_{\mathrm{MKO}}-{K}_{S,2\mathrm{MASS}}$	MK	−0.048	0.00221	⋯	⋯	⋯	⋯	0.006 mag	9.1–13.0 mag
${K}_{\mathrm{MKO}}-{K}_{S,2\mathrm{MASS}}$	MK	−1.526	0.11594	⋯	⋯	⋯	⋯	0.007 mag	13.0–14.2 mag
${K}_{\mathrm{MKO}}-{K}_{S,2\mathrm{MASS}}$	MK	0.054	0.00447	⋯	⋯	⋯	⋯	0.007 mag	14.2–17.0 mag
${K}_{{\rm{H}}2}$ − K	MK	−0.207	0.07241	−0.004693	⋯	⋯	⋯	0.025 mag	9.1–13.1 mag
2MASS Photometric System
$\mathrm{log}({L}_{\mathrm{bol}}/{L}_{\odot })$	${M}_{{K}_{S}}$	−59.877	22.58776	−3.364101	0.2357643	−0.00785837	0.000098821	0.05 dexa	8.8–16.6 mag
$\mathrm{log}({L}_{\mathrm{bol}}/{L}_{\odot })$	MH	−10.426	2.74259	−0.290797	0.0089222	⋯	⋯	0.023 dex	9.2–13.3 mag
${J}_{\mathrm{MKO}}-{J}_{2\mathrm{MASS}}$	${M}_{{K}_{S}}$	−2.747	0.66217	−0.051739	0.0012518	⋯	⋯	0.017 mag	9.1–16.6 mag
${H}_{\mathrm{MKO}}-{H}_{2\mathrm{MASS}}$	${M}_{{K}_{S}}$	−0.754	0.16903	−0.011606	0.0002626	⋯	⋯	0.004 mag	9.1–16.6 mag
${H}_{\mathrm{MKO}}-{H}_{2\mathrm{MASS}}$	MH	−0.050	0.00845	⋯	⋯	⋯	⋯	0.003 mag	9.6–13.4 mag
${K}_{\mathrm{MKO}}-{K}_{S,2\mathrm{MASS}}$	${M}_{{K}_{S}}$	−0.048	0.00219	⋯	⋯	⋯	⋯	0.006 mag	9.1–12.9 mag
${K}_{\mathrm{MKO}}-{K}_{S,2\mathrm{MASS}}$	${M}_{{K}_{S}}$	−1.413	0.10801	⋯	⋯	⋯	⋯	0.026 mag	12.9–14.2 mag
${K}_{\mathrm{MKO}}-{K}_{S,2\mathrm{MASS}}$	${M}_{{K}_{S}}$	0.088	0.00240	⋯	⋯	⋯	⋯	0.007 mag	14.2–17.0 mag
${K}_{{\rm{H}}2}$ − ${K}_{S}$	${M}_{{K}_{S}}$	−0.036	0.03407	−0.002824	⋯	⋯	⋯	0.021 mag	9.1–13.1 mag

Notes. The coefficients are defined as

$$\begin{eqnarray*}&&y=\displaystyle \sum _{i=0}\,{c}_{i}{x}^{i},\end{eqnarray*}$$

where y and x are the quantities listed in the first two columns. For spectral types, M7 corresponds to 7.0 and T5 corresponds to 25.0.

^aThe luminosity relations have significantly different scatter at bright and faint magnitudes. For both, the scatter is 0.04 dex at ≤13.0 mag and 0.07 dex at >13.0 mag.

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Figure 12 also shows literature polynomial relations from Golimowski et al. (2004a) and Filippazzo et al. (2015). Our mass-calibrated relation broadly agrees with this past work, except for the temperature of the L/T transition from Golimowski et al. (2004a), which is systematically high, likely due to the presence of unrecognized binaries in that early work (e.g., Burgasser et al. 2005; Liu et al. 2006). The relation of Filippazzo et al. (2015) is systematically lower than our Lyon relation by ≈90 K, which appears to be due to the fact that they used SM08 model radii. Our SM08 relation agrees very well with Filippazzo et al. (2015), except at the extreme late-type end where we find a 150 K warmer ${T}_{\mathrm{eff}}$ for spectral type T5. This seems to be due to their polynomial not capturing the near-plateau in temperatures through the L/T transition which in our data extends to at least a type of T5. Although our sample does not exactly span the same range in spectral type, the scatter about our polynomial relation (90 K for M7–T5) is somewhat smaller than previous relations spanning M6–T8 (124 K for Golimowski et al. 2004a; 113 K for Filippazzo et al. 2015).

Interestingly, all six T dwarfs in our sample, ranging from T1.5–T5, have effective temperatures that are consistent with each other within the uncertainties, i.e., p(χ²) ≈ 0.5. The weighted average and rms of the model-derived temperatures are 1200 ± 60 K for Lyon and 1190 ± 60 K for SM08. The batch of objects at slightly earlier types are five L6.5–L8.5 dwarfs that also have internally consistent temperatures of 1475 ± 30 K (Lyon) and 1400 ± 25 K (SM08). In comparison, the only object in our sample without a spectral type determination is LHS 2397aB, which has model-derived temperatures of 1520 ± 40 K (Lyon) and 1440 ± 40 K (SM08), consistent with the warmer L6.5–L8.5 dwarfs. This is the first significant sample of objects with individually measured masses spanning the L/T transition. Their model-derived radii combined with our luminosities imply that the spectral type range of L6.5–T5 corresponds to a temperature range of only 200–300 K, with Lyon models favoring the slightly larger temperature range. The narrower ${T}_{\mathrm{eff}}$ range derived from the SM08 models is likely due to the fact that ${L}_{\mathrm{bol}}$ does not drop as steeply through the L/T transition, as discussed in the previous subsection.

Figure 13 shows our sample on the CMD with the data points colored according to their model-derived temperatures. As expected, temperature correlates strongly with the location along the CMD sequence, consistent with temperature being a primary driver of the spectral energy distributions of ultracool dwarfs. Of course, our derived ${T}_{\mathrm{eff}}$ is directly correlated with the plotted absolute magnitudes that are also used to derive ${L}_{\mathrm{bol}}$. However, in the L/T transition, objects have similar absolute magnitudes and yet they span a relatively wide range of temperatures (≈1500 K to ≈1100 K), with objects bluer in J − K having systematically cooler model-derived temperatures.

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The age distribution of the field population provides information about the star formation history of the galaxy and is a key component in studies of substellar mass function in the solar neighborhood (e.g., Burningham et al. 2010; Kirkpatrick et al. 2012). Without directly measured fundamental properties, the age distribution of ultracool dwarfs has previously only been constrained in a statistical sense from kinematic or population synthesis studies (e.g., Burgasser 2004; Allen et al. 2005; Zapatero Osorio et al. 2007; Faherty et al. 2009). However, precise age information is potentially available for brown dwarfs because, unlike stars, they are continuously changing their most easily observable properties, namely luminosity and temperature. The most precise age-dating method within current capabilities is to combine mass and luminosity, two properties that can be measured very accurately as we have shown here and in our past work, and infer an age from evolutionary models (for a review of various brown dwarf age-dating techniques, see Burgasser 2009). Note that, as in all stellar astrophysics, age determinations for brown dwarfs are model dependent, so the accuracy of the derived ages will depend on how well evolutionary models predict substellar luminosity evolution. As we have found in our past work, this is not yet a solved problem, with potential issues at the factor of ≈2 level (e.g., Dupuy et al. 2009b, 2014, 2015b). However, given that our derived ages span nearly two orders of magnitude (∼250 Myr to ∼10 Gyr), such inaccuracies should not have significant influence on the broad trends in our sample.

Figure 14 shows the distribution of system ages for the 10 binaries in our sample with at least one substellar component and thereby a well-determined age. When possible we use SM08 model-derived ages from our total-mass analysis; we only use our individual-mass analysis for the special cases of 2MASS J0700+3157AB, where the secondary is an unresolved binary, and LHS 2397aAB, where the primary is a main-sequence star. We also tested ages derived from Cond models and found that they tended to give slightly (≈10%) older ages. We exclude two systems from our analysis here because their presence in our sample is not independent of their age. HD 130948BC and Gl 569Bab were both discovered in targeted surveys of young stars, and so it would not be correct to include them in this sample of field objects that have different observational selection effects with respect to age. Unlike these two companion systems, Gl 417BC was originally identified on its own in 2MASS and was only later associated with the young star Gl 417 by Kirkpatrick et al. (2001). Likewise, the pre-main-sequence binary LP 349-25AB is not included here because it only has a precise age determination by virtue of its youth.

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The median age of our sample of field brown dwarfs is 1.3 Gyr (2.3 Gyr mean), and the age interval containing 90% of the joint posterior distribution of all 10 systems is 0.4–4.2 Gyr. This age distribution is broadly consistent with or somewhat younger than previous statistical age distributions. From a kinematic analysis of 21 L and T dwarfs, Zapatero Osorio et al. (2007) found a statistical age of ${1.2}_{-0.7}^{+1.1}$ Gyr, in good agreement with our age distribution. (Their sample includes just one of our age-dated systems, 2MASS J1728+3948AB.) In contrast, Faherty et al. (2009) found an older statistical age range from a kinematic analysis of proper motions for a much larger sample of 184 L0–T8 dwarfs, ranging from 3–8 Gyr for the whole sample, or 2–4 Gyr excluding high tangential velocity (>100 km s⁻¹) objects. (None of our systems have such high tangential velocities.) The Faherty et al. (2009) results are thus systematically older than, but marginally consistent with, our age distribution.

Allen et al. (2005) found that brown dwarfs with spectral types ranging from L5 to early-T are expected to have a mean age of 3 Gyr, given a uniform prior on age from 0–10 Gyr and a nominal mass function of ${\rm{\Psi }}(m)\propto {m}^{-0.8}$. The fact that we find a younger mean age could imply that the solar neighborhood comprises objects with a slightly younger age distribution. To investigate this possible discrepancy we performed a population synthesis simulation using a power-law mass function that is consistent with recent works (${\rm{\Psi }}(m)\propto {m}^{0.5};$ Kirkpatrick et al. 2012; Burningham et al. 2013) for a range in mass of 30–70 ${M}_{\mathrm{Jup}}$, i.e., the lowest to highest masses covered by our age-dated sample. Rather than assume a constant input age distribution, we adopted the Besançon model for the solar neighborhood that assumes a constant star formation rate and accounts for Galactic dynamics in a self-consistent way (Robin et al. 2003).¹³ Because the Sun lies near the Galactic midplane, dynamical heating skews the age distribution toward younger ages as older stars are preferentially scattered away from the midplane over time. In our simulation, we assume that ages are distributed uniformly among each of the Besançon model age bins, where each bin contains a fraction of the total population as follows: 20.0% for 0.15–1 Gyr, 16.1% for 1–2 Gyr, 11.9% for 2–3 Gyr, 16.6% for 3–5 Gyr, 12.9% for 5–7 Gyr, and 14.4% for 7–10 Gyr. We drew random ages according to this distribution, restricting our population synthesis simulation to 0.3–10 Gyr in order to accurately represent our observed sample.

After assigning masses and ages to each simulated object, we interpolated ${T}_{\mathrm{eff}}$ from the SM08 models, keeping only simulated objects that fall within the temperature range of our sample (1100–2100 K). As a check, the fractional breakdown in spectral types between ≤L6, L6.5–L8.5, and L9–T5 dwarfs was 0.22, 0.36, and 0.42, very similar to the actual breakdown of objects in our age-dated sample (0.28, 0.33, 0.39). Also note that by performing our simulation with the same evolutionary models used to derive ages for our sample, the two age distributions are self-consistent by construction. In other words, the simulation output is directly comparable to our age distribution, even if the models were to have large systematic uncertainties in the absolute ages.

The age distribution resulting from our simple population synthesis simulation is shown in Figure 14. It has a median of 1.7 Gyr, a mean of 2.3 Gyr, and the interval 0.3–5.0 Gyr contains 90% of the distribution. The general shape of the distribution is quite similar to our observations, piling up at younger ages and tailing off at older ages. We checked if the input mass function might change this result, but using α = 0 or −1 instead of −0.5 only changed the final median age by ≈5%. The simulation predicts that we would have found ≈1 old system (>5 Gyr) in our sample of 10 binaries, and the fact that none of our sample is definitively that old is the main cause of the slight discrepancy between the population synthesis and our observations. Overall, however, our age distribution is remarkably consistent with our input assumption of a constant star formation rate.

The simulated population accounts for the main selection effects in our sample, namely a limited spectral type range and the fact that we can only age-date brown dwarfs and not stars (i.e., objects with mass ≲70 ${M}_{\mathrm{Jup}}$). The spectral type selection should bias our sample against old ages, and indeed the simulation indicates that we are the most complete at young ages. There is one remaining selection effect that is difficult to quantify, the fact that the youngest binaries at a given spectral type will have the lowest masses, and lower system masses correspond to longer orbital periods for a given semimajor axis range. Our dynamical mass sample relies on robust orbit determinations and thus suffers incompleteness at long periods (and thereby low masses and young ages) because there has been insufficient time to constrain the longest period orbits. (See Section 7.8 for a detailed discussion of this selection effect.) The fact that our sample is biased in this way against young ages, and yet we still find an age distribution skewed slightly younger than our simulation suggests that this selection effect has only a marginal influence.

Overall, our population synthesis simulation demonstrates that our observed age distribution is consistent with a constant star formation rate in the Galactic disk. We find that the age distribution of M7–T5 dwarfs in the solar neighborhood is somewhat younger than in previous works, but we determine that this can be explained by accounting for dynamical heating in the disk that boosts the scale height of older stars, removing them from the local volume. As a test, we performed a second simulation with a uniform age input (i.e., constant star formation rate without dynamical heating), and this skewed the output median age older by almost a factor of two. In the future it will be possible to extend to older ages this empirical determination of the age distribution in the solar neighborhood with dynamical mass measurements for even cooler brown dwarfs (e.g., Dupuy et al. 2015a).

We have previously examined the eccentricities of ultracool binaries and the implications for formation models in Dupuy & Liu (2011). The sample we used previously included eleven visual binaries, four spectroscopic binaries, and one eclipsing binary, and we had to contend with potential selection effects in this sample due to discovery bias (eccentric visual binaries are easier to discover with poor angular resolution), our own bias in selecting which binaries to monitor, and whether eccentricity might make some binaries more difficult to yield orbit determinations. Our new sample is not only much larger (10 new orbits), but we are also in a better position to address issues regarding observer and orbit-fitting biases. This is because we present here our entire monitoring sample with orbit fits for all binaries. The only selection is therefore based on the quality of the resulting fits, and this allows us to robustly quantify the completeness of our sample.

As described in detail in Section 2, our input sample of 31 binaries is, to the best of our knowledge, complete for binaries with projected separations at discovery of ≤6 au, given our other non-orbit related selection criteria (spectral types M6.5–T5.5, d ≤ 40 pc, observable from Maunakea with Keck LGS AO). Figure 15 shows all orbit-fitting results for our sample, including binaries with marginally or poorly determined orbits, as well as visual binary orbits from the literature. As expected, the binaries with the widest separations at discovery have turned out to have longer orbital periods. Because we are hampered by the available observational time baseline for longer period binaries, many of these do not have well-determined orbits. Only one binary with P > 30 years has a well-determined orbit (2MASS J1728+3948AB); all the rest are marginal or poor. The only binary at P < 30 years that does not have a well-determined orbit is SDSS J0926+5847AB (e = 0.00–0.58 at 2σ). This is a pathological case where the orbit is very close to being viewed edge-on, and the phase coverage from our three-year HST program does not allow us to uniquely determine the orbit. It is likely that if SDSS J0926+5847AB were observable with Keck LGS AO, then we would have better phase coverage over a longer time baseline that would enable an orbit fit. Therefore, we conclude that for periods <30 years, our sample of orbit determinations is effectively complete, including 22 binaries from our sample and 3 binaries from the literature.

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A complete sample can still be biased, with the key effect here being "discovery bias," which is explored in detail in Section 3.1.1 of Dupuy & Liu (2011). Briefly, every imaging survey has an inner working angle (IWA), inside of which binary companions are not detectable. Given a binary with semimajor axis a, the projected separation will sometimes be inside of this limit, depending on the inclination, eccentricity, and epoch of observation. When IWA ≪ a, the probability of detecting any binary approaches unity. When IWA > a, circular binaries are completely undetectable while more eccentric binaries are more favorable to detect because the apoastron distance is $a\times (1+e)$ (see Figure 1 of Dupuy & Liu 2011). By excluding binaries discovered very near the IWA of a survey, we can minimize the impact of this discovery bias. Figure 16 shows the ratio of IWA/a for binaries in our sample. The highest IWA/a ratios are 1.58 and 1.24 for 2MASS J1047+4026AB and LP 415-20AB, respectively, and indeed both of these binaries are quite eccentric (0.7485 ± 0.0013 and ${0.706}_{-0.012}^{+0.011}$). The next highest are IWA/a = 0.88 for 2MASS J0920+3517AB ($e={0.180}_{-0.007}^{+0.006}$) and IWA/a = 0.69 for LP 349-25AB ($e={0.0468}_{-0.0018}^{+0.0019}$). Based on the simulations of Dupuy & Liu (2011), we adopt a cutoff of IWA/a < 0.75 to mitigate discovery bias in our sample, and this includes 2MASS J1047+4026AB, LP 415-20AB, and the more marginal case of 2MASS J0920+3517AB. Eccentricities for our entire sample, along with the three other published visual binary orbits, are shown in Figure 17 as a function of semimajor axis.

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After making the cuts described above, the final complete de-biased sample comprises 22 ultracool visual binaries. In addition, we examine published orbits for various unresolved binaries: three astrometric orbits (Sahlmann et al. 2015a, 2015b; Koren et al. 2016), three spectroscopic orbits (Basri & Martín 1999; Joergens et al. 2010; Burgasser et al. 2016), and one double-lined eclipsing orbit for a very young brown dwarf binary in Orion (Stassun et al. 2006). Figure 18 shows all of these results, spanning more than three orders of magnitude in period. The unresolved binaries all have modest eccentricities (0.2–0.6), whereas our sample has a number of very eccentric (>0.7) and nearly circular (0.0–0.1) systems. The lack of low eccentricities (<0.2) in the seven unresolved binaries is the most striking, as nearly half (10 of 22) of the visual binaries have e < 0.2. More short-period binaries are needed to determine the significance of this apparent difference, given the current small sample with short periods.

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Figure 19 shows the eccentricity distribution of our de-biased visual binary sample. The abundance of low eccentricity orbits is the clearest result, reinforcing the conclusion of Dupuy & Liu (2011) that very low-mass binaries are very inconsistent with the high eccentricity orbits found in simulations from Stamatellos & Whitworth (2009) but consistent with the more modest eccentricities predicted from the simulations of Bate (2009). In fact, our larger sample here is less consistent with the Bate (2009) results that do not predict as many nearly circular orbits. For both sets of simulations, it is possible that the influence of gas over timescales longer than the simulated durations would damp eccentricities further and bring the simulations into better agreement with our observations.

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Compared to the extensive orbital information available for solar-type binaries spanning ≈6 orders of magnitude in period (e.g., Duquennoy & Mayor 1991; Raghavan et al. 2010), our visual binary sample covers a relatively narrow range of periods (≲1 order of magnitude). However, one striking feature of the recent large Raghavan et al. (2010) compilation of solar-type multiple systems (127 eccentricities) is that for a wide range of periods (∼10²–10⁶ days), there are no binary orbits with e < 0.1 and only three orbits in triple or quadruple systems with e < 0.1. In contrast, our de-biased visual binary sample has 5 out of 22 orbits with e < 0.1. Two of these are orbits in triple systems, where one is an "inner" orbit (i.e., the orbit of the tighter pair in the hierarchical triple; HD 130948BC) and one is an "outer" outer (2MASS J0700+3157AB). The other three all seem to be binaries with no other companions (LP 349-25AB, 2MASS J1534−2952AB, and 2MASS J2206−2047AB). This discrepancy between the samples of orbits of very low-mass and solar-type objects was first noted in Dupuy & Liu (2011), and it is even stronger now after roughly doubling the sample size of visual binaries. For example, the Fisher exact test comparing the number of binaries above and below e = 0.1 in our sample to Raghavan et al. (2010) gives a p-value of 0.0019, implying the two samples are inconsistent at 3.1σ. This difference in low-e orbits could be due to differences in initial conditions (i.e., binary formation is different at very low masses) and/or early evolution (i.e., different physical processes at work after formation). In the latter case, if dissipative gas disks are longer lived for very low-mass objects than solar-type stars, this could potentially result in a higher fraction of nearly circular orbits even if the initial conditions are the same for both samples.

As described in Section 4.1, we have excluded some binaries from our analysis because of their uncertain orbits. This is largely due to poor orbital coverage given their apparently long orbital periods relative to our observational time baseline. Since our sample was selected to have the smallest semimajor axes possible, some of these binaries are likely to be the lowest mass systems in our sample. Indeed, if we take our orbit determinations at face value, then 2MASS J0850+1057AB, SDSS J1021−0304AB, SDSS J1534+1615AB, and SDSSJ2052−1609AB all have lower total masses than the lowest mass object used in our analysis (SDSS J0423−0414AB, ${M}_{\mathrm{tot}}=83\pm 3$ ${M}_{\mathrm{Jup}}$).

2MASS J0850+1057AB (L6.5+L8.5) has a total mass of only 54 ± 8 ${M}_{\mathrm{Jup}}$, and thus perhaps component masses of 20–30 ${M}_{\mathrm{Jup}}$. Given the component luminosities, this would correspond to an age of 150–400 Myr according to both the Lyon and SM08 models. The 2MASS J0850+1057 system has previously been suggested to be young based on having an M-dwarf companion (NLTT 20346; 413 or 8000 au away) with a young X-ray activity age (Faherty et al. 2011) and based on 2MASS J0850+1057A being bright at the J-band for its spectral type (Burgasser et al. 2011). However, the physical association of NLTT 20346 with the 2MASS J0850+1057 system is questionable (see Section 6.4 of Dupuy & Liu 2012), and the discovery and characterization of young late-L dwarfs indicate that they tend to be fainter at the J-band, not brighter (e.g., Faherty et al. 2012; Liu et al. 2013; Liu et al. 2016). Our improved proper motion measurement for the 2MASS J0850+1057 system is consistent with our previous findings, discrepant by 45 mas yr⁻¹ (6.3σ) with the Faherty et al. (2011) proper motion for NLTT 20346, implying that the wide pair is not likely to be physically associated. Burgasser et al. (2011) also suggested that 2MASS J0850+1057A could be an unresolved binary based on its unusually bright J- and K-band absolute magnitudes given its spectral type of L7, making this a triple system of ultracool dwarfs. Our distance (31.8 ± 0.6 pc instead of 38 ± 6 pc) and spectral type of L6.5 ± 1.0 give absolute magnitudes comparable to other L5–L7 dwarfs (see Table 15 of Dupuy & Liu 2012), reducing the reason to suspect 2MASS J0850+1057A is an unresolved binary. Moreover, if our low total mass is accurate, then the component masses (if a triple system) would be 10–15 ${M}_{\mathrm{Jup}}$, making the system even younger. This seems unlikely given that the integrated-light spectrum lacks spectral similarities to other comparably young late-L dwarfs (e.g., Liu et al. 2013; Gizis et al. 2015).

SDSS J1021−0304AB (T0+T5, ${M}_{\mathrm{tot}}={52}_{-7}^{+6}$ ${M}_{\mathrm{Jup}}$) and SDSS J1534+1615AB (T0+T5.5, ${M}_{\mathrm{tot}}={46}_{-7}^{+6}$ ${M}_{\mathrm{Jup}}$) have the most precise masses of any of the systems with poorly determined orbits. For plausible mass ratios, the implied component masses of these binaries would be 20–30 ${M}_{\mathrm{Jup}}$. With orbital period posteriors of ${86}_{-17}^{+13}$ years and ${58}_{-24}^{+39}$ years, respectively, it is unlikely that we will achieve significant improvement in their orbit determinations in the near future. SDSS J2052−1609AB (L8.5+T1.5, ${M}_{\mathrm{tot}}={69}_{-20}^{+14}$ ${M}_{\mathrm{Jup}}$) has a very uncertain total mass but the best chance of yielding a robust orbit in the near term, as its derived period is ${44}_{-14}^{+10}$ years and our observations already cover 20% of this.

In order to report complete photometry across both MKO and 2MASS photometric systems for our sample binaries, we compute photometric system conversions based solely on K-band absolute magnitude. We used our compilation of ultracool dwarf parallaxes available online,¹⁴ selecting only single objects with a published spectrum, a parallax error of <15%, and a "null" object flag, meaning that there is nothing unusual about the object. We use synthesized 2MASS/MKO photometric conversions from Dupuy & Liu (2012) for each of the 46 objects in each of the J-, H-, and K-bands. For the J- and H-bands, we found that a third-order polynomial provided a good fit to the data. The K-band conversion undergoes a steep jump between M_K ≈ 13–14 mag that cannot be well-approximated by a single polynomial. Therefore, we computed a three-part piecewise linear fit to the K-band conversion as a function of M_K. In addition to these relations as a function of M_K, we also computed an H-band conversion as a function of M_H. The polynomial coefficients are given in Table 14 and the fits are displayed in Figure 20. The typical rms about the fits ranges from 0.003 mag for the H-band to 0.015 mag for the J-band.

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Determining mass ratios from photocenter motion requires knowing the flux ratios in the imaging bandpass. Our absolute astrometry for the brightest targets observed with CFHT/WIRCam was obtained in a narrow K-band filter, ${K}_{{\rm{H}}2}$, centered on 2.122 μm with a bandwidth of 0.032 μm. We do not have resolved photometry for these binaries in this specialized filter, so we must synthesize it from our broadband K photometry in order to determine the flux ratios of our binaries in our CFHT imaging.

We used the same sample of objects selected from our online parallax compilation as described in the previous section. We used the filter curve provided by CFHT to synthesize ${K}_{{\rm{H}}2}$ photometry,¹⁵ and we assume that the difference between this scan done in the lab and the transmission at colder operational temperatures is negligible for our purposes. We computed ${K}_{{\rm{H}}2}$ − K and ${K}_{{\rm{H}}2}$ − ${K}_{S}$ colors for these objects and found a smooth relation for objects with M_K < 13.1 mag, corresponding roughly to spectral types <L9. (The smooth relations break down for later types as the ${K}_{{\rm{H}}2}$ filter samples part of the K-band flux peak of T dwarfs while their broadband K flux plummets.) We fit a second-order polynomial as a function of M_K for 25 objects. The coefficients are given in Table 14 and the fits are displayed in Figure 20. The rms about these fits were 0.025 mag for K and 0.021 mag for ${K}_{S}$.

Determining the component luminosities using our resolved photometry is a critical part of our model tests. We have used the large sample of luminosities from Filippazzo et al. (2015) to derive polynomial relations between luminosity and absolute magnitude. We used only objects with parallactic distances, nominal ages ≳0.5 Gyr, and normal properties (i.e., no subdwarfs, young objects, or objects dubbed peculiar in their spectral type). We supplemented the observed magnitudes reported by Filippazzo et al. (2015) by applying our 2MASS/MKO conversions from above. For any object with either a 2MASS or MKO measurement, we derived the complementary magnitude using the K-band absolute magnitude. For objects with both 2MASS and MKO measurements, we used the more precise magnitude to derive the complementary magnitude, supplanting the directly observed value. This results in a sample of 126 normal field objects with luminosities and K-band absolute magnitudes. (Since our 2MASS/MKO relations are only valid between M_K = 9.1–17.0 mag, there are a few objects with only 2MASS or MKO photometry that changes the input sample slightly between 2MASS and MKO.) A subset of 76 of these objects had magnitudes of M_H < 13.3 mag, which we used in our H-band relations.

We fit luminosity as a function of K-band absolute magnitude with a fifth-order polynomial, since there was a clear structure in the residuals using only a third-order polynomial. For the more limited H-band magnitude range, a third-order polynomial was sufficient. In the K-band, it was visually apparent that the scatter at brighter magnitudes was less than at fainter magnitudes. The fact that the nominal errors on the magnitudes and luminosities are not significantly larger for the fainter objects implies that this increased scatter is due to either unquantified systematic errors in the luminosities of the lower luminosity objects or real astrophysical scatter in the relationship between the K-band absolute magnitude and luminosity. Regardless, we quantify this effect by separately computing the rms scatter at the bright end (M_K ≤ 13.0 mag; rms of 0.04 dex for 90 objects) and faint end (rms of 0.07 dex for 36 objects). The coefficients for all relations are given in Table 14, and the fits and residuals are displayed in Figure 21.

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